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How does LINQ's OrderBy jive with MoveNext?

This thread says that LINQ's OrderBy uses Quicksort. I'm struggling how that makes sense given that OrderBy returns an IEnumerable.
Let's take the following piece of code for example.
int[] arr = new int[] { 1, -1, 0, 60, -1032, 9, 1 };
var ordered = arr.OrderBy(i => i);
foreach(int i in ordered)
Console.WriteLine(i);
The loop is the equivalent of
var mover = ordered.GetEnumerator();
while(mover.MoveNext())
Console.WriteLine(mover.Current);
The MoveNext() returns the next smallest element. The way that LINQ works, unless you "cash out" of the query by use ToList() or similar, there are not supposed to be any intermediate lists created, so each time you call MoveNext() the IEnumerator finds the next smallest element. That doesn't make sense because during the execution of Quicksort there is no concept of a current smallest and next smallest element.
Where is the flaw in my thinking here?

the way that LINQ works, unless you "cash out" of the query by use ToList() or similar, there are not supposed to be any intermediate lists created
This statement is false. The flaw in your thinking is that you believe a false statement.
The LINQ to Objects implementation is smart about deferring work when possible at a reasonable cost. As you correctly note, it is not possible in the case of sorting. OrderBy produces as its result an object which, when MoveNext is called, enumerates the entire source sequence, generates the sorted list in memory and then enumerates the sorted list.
Similarly, joining and grouping also must enumerate the whole sequence before the first element is enumerated. (Logically, a join is just a cross product with a filter, and the work could be spread out over each MoveNext() but that would be inefficient; for practicality, a lookup table is built. It is educational to work out the asymptotic space vs time tradeoff; give it a shot.)
The source code is available; I encourage you to read it if you have questions about the implementation. Or check out Jon's "edulinq" series.

There's a great answer already, but to add a few things:
Enumerating the results of OrderBy() obviously can't yield an element until it has processed all elements because not until it has seen the last input element can it know that that last element seen isn't the first it must yield. It also must work on sources that can't be repeated or which will give different results each time. As such even if some sort of zeal meant the developers wanted to find the nth element anew each cycle, buffering is a logical requirement.
The quicksort is lazy in two regards though. One is that rather than sort the elements to return based on the keys from the delegate passed to the method, it sorts a mapping:
Buffer all the elements.
Get the keys. Note that this means the delegate is run only once per element. Among other things it means that non-pure keyselectors won't cause problems.
Get a map of numbers from 0 to n.
Sort the map.
Enumerate through the map, yielding the associated element each time.
So there is a sort of laziness in the final sorting of elements. This is significant in cases where moving elements is expensive (large value types).
There is of course also laziness in that none of the above is done until after the first attempt to enumerate, so until you call MoveNext() the first time, it won't have happened.
In .NET Core there is further laziness building on that, depending on what you then do with the results of OrderBy. Since OrderBy contains information about how to sort rather than the sorted buffer, the class returned by OrderBy can do something else with that information other than quicksorting:
The most obvious is ThenBy which all implementations do. When you call ThenBy or ThenByDescending you get a new similar class with different information about how to sort, and the sort the OrderBy result could have done probably never will.
First() and Last() don't need to sort at all. Logically source.OrderBy(del).First() is a variant of source.Min() where del contains the information to determine what defines "less than" for that Min(). Therefore if you call First() on the results of an OrderBy() that's exactly what is done. The laziness of OrderBy allows it to do this instead of quicksort. (Which means O(n) time complexity and O(1) space complexity instead of O(n log n) and O(n) respectively).
Skip() and Take() define a subsequence of a sequence which with OrderBy must conceptually happen after that sort. But since they are lazy too what can be returned is an object that knows; how to sort, how many to skip, how many to take. As such partial quicksort can be used so that the source need only be partially sorted: If a partition is outside of the range that will be returned then there's no point sorting it.
ElementAt() places more of a burden than First() or Last() but again doesn't require a full quicksort. Quickselect can be used to find just one result; if you're looking for the 3rd element and you've partitioned a set of 200 elements around the 90th element then you only need to look further in the first partition and can ignore the second partition from now on. Best-case and average-case time complexity is O(n).
The above can be combined, so e.g. .Skip(10).First() is equivalent to ElementAt(10) and can be treated as such.
All of these exceptions to getting the entire buffer and sorting it all have one thing in common: They were all implemented after identifying a way in which the correct result can be returned after making the computer do less work*. That new [] {1, 2, 3, 4}.Where(i => i % 2 == 0) will yield the 2 before it has seen the 4 (or even the 3 it won't yield comes from the same general principle. It just comes at it more easily (though there are still specialised variants of Where() results behind the scenes to provide other optimisations).
But note that Enumerable.Range(1, 10000).Where(i => i >= 10000) scans through 9999 elements to yield that first. Really it's not all that different to OrderBy's buffering; they're both bringing you the next result as quickly as they can†, and what differs is just what that means.
*And also identifying that the effort to detect and make use of the features of a particular case are worth it. E.g. many aggregate calls like Sum() can be optimised of the results of OrderBy by skipping the ordering completely. But this can generally be realised by the caller and they can just leave out the OrderBy so while adding that would make most calls to Sum() slightly slower to make that case much faster the case that benefits shouldn't really be happening anyway.
†Well, pretty much as quickly. It would be possible to get the first results back more quickly than OrderBy does—when you've got the left most part of a sequence sorted start giving out results—but that comes at a cost that would affect the later results so the trade-off isn't necessarily that doing that would be better.

Using AsSequential in order to preserve order

I am looking at this code
var numbers = Enumerable.Range(0, 20);
var parallelResult = numbers.AsParallel().AsOrdered()
.Where(i => i % 2 == 0).AsSequential();
foreach (int i in parallelResult.Take(5))
Console.WriteLine(i);
The AsSequential() is supposed to make the resulting array sorted. Actually it is sorted after its execution, but if I remove the call to AsSequential(), it is still sorted (since AsOrdered()) is called.
What is the difference between the two?

AsSequential is just meant to stop any further parallel execution - hence the name. I'm not sure where you got the idea that it's "supposed to make the resulting array sorted". The documentation is pretty clear:
Converts a ParallelQuery into an IEnumerable to force sequential evaluation of the query.
As you say, AsOrdered ensures ordering (for that particular sequence).

I know that this was asked over a year old but here are my two cents.
In the example exposed, i think it uses AsSequential so that the next query operator (in this case the Take operator) it is execute sequentially.
However the Take operator prevent a query from being parallelized, unless the source elements are in their original indexing position, so that is why even when you remove the AsSequential operator, the result is still sorted.

Does LINQ cache computed values?

Suppose I have the following code:
var X = XElement.Parse (#"
<ROOT>
<MUL v='2' />
<MUL v='3' />
</ROOT>
");
Enumerable.Range (1, 100)
.Select (s => X.Elements ()
.Select (t => Int32.Parse (t.Attribute ("v").Value))
.Aggregate (s, (t, u) => t * u)
)
.ToList ()
.ForEach (s => Console.WriteLine (s));
What is the .NET runtime actually doing here? Is it parsing and converting the attributes to integers each of the 100 times, or is it smart enough to figure out that it should cache the parsed values and not repeat the computation for each element in the range?
Moreover, how would I go about figuring out something like this myself?
Thanks in advance for your help.

LINQ and IEnumerable<T> is pull based. This means that the predicates and actions that are part of the LINQ statement in general are not executed until values are pulled. Furthermore the predicates and actions will execute each time values are pulled (e.g. there is no secret caching going on).
Pulling from an IEnumerable<T> is done by the foreach statement which really is syntactic sugar for getting an enumerator by calling IEnumerable<T>.GetEnumerator() and repeatedly calling IEnumerator<T>.MoveNext() to pull the values.
LINQ operators like ToList(), ToArray(), ToDictionary() and ToLookup() wraps a foreach statement so these methods will do a pull. The same can be said about operators like Aggregate(), Count() and First(). These methods have in common that they produce a single result that has to be created by executing a foreach statement.
Many LINQ operators produce a new IEnumerable<T> sequence. When an element is pulled from the resulting sequence the operator pulls one or more elements from the source sequence. The Select() operator is the most obvious example but other examples are SelectMany(), Where(), Concat(), Union(), Distinct(), Skip() and Take(). These operators don't do any caching. When then N'th element is pulled from a Select() it pulls the N´th element from the source sequence, applies the projection using the action supplied and returns it. Nothing secret going on here.
Other LINQ operators also produce new IEnumerable<T> sequences but they are implemented by actually pulling the entire source sequence, doing their job and then producing a new sequence. These methods include Reverse(), OrderBy() and GroupBy(). However, the pull done by the operator is only performed when the operator itself is pulled meaning that you still need a foreach loop "at the end" of the LINQ statement before anything is executed. You could argue that these operators use a cache because they immediately pull the entire source sequence. However, this cache is built each time the operator is iterated so it is really an implementation detail and not something that will magically detect that you are applying the same OrderBy() operation multiple times to the same sequence.
In your example the ToList() will do a pull. The action in the outer Select will execute 100 times. Each time this action is executed the Aggregate() will do another pull that will parse the XML attributes. In total your code will call Int32.Parse() 200 times.
You can improve this by pulling the attributes once instead of on each iteration:
var X = XElement.Parse (#"
<ROOT>
<MUL v='2' />
<MUL v='3' />
</ROOT>
")
.Elements ()
.Select (t => Int32.Parse (t.Attribute ("v").Value))
.ToList ();
Enumerable.Range (1, 100)
.Select (s => x.Aggregate (s, (t, u) => t * u))
.ToList ()
.ForEach (s => Console.WriteLine (s));
Now Int32.Parse() is only called 2 times. However, the cost is that a list of attribute values have to be allocated, stored and eventually garbage collected. (Not a big concern when the list contains two elements.)
Note that if you forget the first ToList() that pulls the attributes the code will still run but with the exact same performance characteristics as the original code. No space is used to store the attributes but they are parsed on each iteration.

It has been a while since I dug through this code but, IIRC, the way Select works is to simply cache the Func you supply it and run it on the source collection one at a time. So, for each element in the outer range, it will run the inner Select/Aggregate sequence as if it were the first time. There isn't any built-in caching going on -- you would have to implement that yourself in the expressions.
If you wanted to figure this out yourself, you've got three basic options:
Compile the code and use ildasm to view the IL; it's the most accurate but, especially with lambdas and closures, what you get from IL may look nothing like what you put into the C# compiler.
Use something like dotPeek to decompile System.Linq.dll into C#; again, what you get out of these kinds of tools may only approximately resemble the original source code, but at least it will be C# (and dotPeek in particular does a pretty good job, and is free.)
My personal preference - download the .NET 4.0 Reference Source and look for yourself; this is what it's for :) You have to just trust MS that the reference source matches the actual source used to produce the binaries, but I don't see any good reason to doubt them.
As pointed out by #AllonGuralnek you can set breakpoints on specific lambda expressions within a single line; put your cursor somewhere inside the body of the lambda and press F9 and it will breakpoint just the lambda. (If you do it wrong, it will highlight the entire line in the breakpoint color; if you do it right, it will just highlight the lambda.)

C# Linq Where(expression).FirstorDefault() vs .FirstOrDefault(expression)

What is the difference between these two Linq queries:
var result = ResultLists().Where( c=> c.code == "abc").FirstOrDefault();
// vs.
var result = ResultLists().FirstOrDefault( c => c.code == "abc");
Are the semantics exactly the same?
Iff sematically equal, does the predicate form of FirstOrDefault offer any theoretical or practical performance benefit over Where() plus plain FirstOrDefault()?

Either is fine.
They both run lazily - if the source list has a million items, but the tenth item matches then both will only iterate 10 items from the source.
Performance should be almost identical and any difference would be totally insignificant.

The second one. All other things being equal, the iterator in the second case can stop as soon as it finds a match, where the first one must find all that match, and then pick the first of those.

Nice discussion, all the above answers are correct.
I didn't run any performance test, whereas on the bases of my experience FirstOrDefault() sometimes faster and optimize as compare to Where().FirstOrDefault().
I recently fixed the memory overflow/performance issue ("neural-network algorithm") and fix was changing Where(x->...).FirstOrDefault() to simply FirstOrDefault(x->..).
I was ignoring the editor's recommendation to change Where(x->...).FirstOrDefault() to simply FirstOrDefault(x->..).
So I believe the correct answer to the above question is
The second option is the best approach in all cases

Where is actually a deferred execution - it means, the evaluation of an expression is delayed until its realized value is actually required. It greatly improves performance by avoiding unnecessary execution.
Where looks kind of like this, and returns a new IEnumerable
foreach (var item in enumerable)
{
if (condition)
{
yield return item;
}
}
FirstOrDefault() returns <T> and not throw any exception or return null when there is no result

What's the difference between these two statements?

List<int> result1 =
(from number in list where number < 3 select number).ToList();
List<int> result2 = list.Where(n => n<3).ToList();
What's the difference between these two different statements?

The first notation is usually called "query syntax", the second one "method syntax" (or dot notation, or lambda syntax) - both are compiled down to exactly the same code, but as already mentioned usually one of the two is more succinct, for most scenarios this is the dot notation but especially for joining or grouping over multiple enumerations query syntax really shines.
Also check out LINQ Query Syntax versus Method Syntax (C#):
Most queries in the introductory LINQ
documentation are written as query
expressions by using the declarative
query syntax introduced in C# 3.0.
However, the .NET common language
runtime (CLR) has no notion of query
syntax in itself. Therefore, at
compile time, query expressions are
translated to something that the CLR
does understand: method calls. These
methods are called the standard query
operators, and they have names such as
Where, Select, GroupBy, Join, Max,
Average, and so on. You can call them
directly by using method syntax
instead of query syntax.
In general, we recommend query syntax
because it is usually simpler and more
readable; however there is no semantic
difference between method syntax and
query syntax.

Nothing.
The first one uses LINQ notation, while the second one uses extension method notation -- they both do the same thing.
Use whatever looks more pleasing to you. :)

There is no difference. One is just a language extension that looks similar to SQL instead of using delegates to achieve the same result.

You notice already the first is LINQ notation and the second one uses extension method with lambda. Use the second for less code maintainance. but if you think the similarity of internal code or performance, simply use stop watch and run this code 100000 times and choose the fastest one. If the compiled code is similar, you will get the time almost the same.