Getting hash of a list of strings regardless of order

An alternative to sorting the string lists would be to get the hash codes of the strings and then sort the hash codes. (Comparing ints is less expensive than comparing strings.) You can then use an algorithm to merge the hash codes that (hopefully) gives a better distribution.

Example:

GetHashCodeOfList<T>(IEnumerable<T> list) {
   List<int> codes = new List<int>();
   foreach (T item in list) {
      codes.Add(item.GetHashCode());
   }
   codes.Sort();
   int hash = 0;
   foreach (int code in codes) {
      unchecked {
         hash *= 251; // multiply by a prime number
         hash += code; // add next hash code
      }
   }
   return hash;
}

There are various different approaches here the under two main categories, each typically with their own benefits and disadvantages, in terms of effectiveness and performance. It is probably best to choose the simplest algorithm for whatever application and only use the more complex variants if necessary for whatever situation.

Note that these examples use EqualityComparer<T>.Default since that will deal with null elements cleanly. You could do better than zero for null if desired. If T is constrained to struct it is also unnecessary. You can hoist the EqualityComparer<T>.Default lookup out of the function if so desired.

Commutative Operations

If you use operations on the hashcodes of the individual entries which are commutative then this will lead to the same end result regardless of order.

There are several obvious options on numbers:

XOR

public static int GetOrderIndependentHashCode<T>(IEnumerable<T> source)
{
    int hash = 0;
    foreach (T element in source)
    {
        hash = hash ^ EqualityComparer<T>.Default.GetHashCode(element);
    }
    return hash;
}

One downside of that is that the hash for { "x", "x" } is the same as the hash for { "y", "y" }. If that's not a problem for your situation though, it's probably the simplest solution.

Addition

public static int GetOrderIndependentHashCode<T>(IEnumerable<T> source)
{
    int hash = 0;
    foreach (T element in source)
    {
        hash = unchecked (hash + 
            EqualityComparer<T>.Default.GetHashCode(element));
    }
    return hash;
}

Overflow is fine here, hence the explicit unchecked context.

There are still some nasty cases (e.g. {1, -1} and {2, -2}, but it's more likely to be okay, particularly with strings. In the case of lists that may contain such integers, you could always implement a custom hashing function (perhaps one that takes the index of recurrence of the specific value as a parameter and returns a unique hash code accordingly).

Here is an example of such an algorithm that gets around the aforementioned problem in a fairly efficient manner. It also has the benefit of greatly increasing the distribution of the hash codes generated (see the article linked at the end for some explanation). A mathematical/statistical analysis of exactly how this algorithm produces "better" hash codes would be quite advanced, but testing it across a large range of input values and plotting the results should verify it well enough.

public static int GetOrderIndependentHashCode<T>(IEnumerable<T> source)
{
    int hash = 0;
    int curHash;
    int bitOffset = 0;
    // Stores number of occurences so far of each value.
    var valueCounts = new Dictionary<T, int>();

    foreach (T element in source)
    {
        curHash = EqualityComparer<T>.Default.GetHashCode(element);
        if (valueCounts.TryGetValue(element, out bitOffset))
            valueCounts[element] = bitOffset + 1;
        else
            valueCounts.Add(element, bitOffset);

        // The current hash code is shifted (with wrapping) one bit
        // further left on each successive recurrence of a certain
        // value to widen the distribution.
        // 37 is an arbitrary low prime number that helps the
        // algorithm to smooth out the distribution.
        hash = unchecked(hash + ((curHash << bitOffset) |
            (curHash >> (32 - bitOffset))) * 37);
    }

    return hash;
}

Multiplication

Which has few if benefits over addition: small numbers and a mix of positive and negative numbers they may lead to a better distribution of hash bits. As a negative to offset this "1" becomes a useless entry contributing nothing and any zero element results in a zero. You can special-case zero not to cause this major flaw.

public static int GetOrderIndependentHashCode<T>(IEnumerable<T> source)
{
    int hash = 17;
    foreach (T element in source)
    {
        int h = EqualityComparer<T>.Default.GetHashCode(element);
        if (h != 0)
            hash = unchecked (hash * h);
    }
    return hash;
}

Order first

The other core approach is to enforce some ordering first, then use any hash combination function you like. The ordering itself is immaterial so long as it is consistent.

public static int GetOrderIndependentHashCode<T>(IEnumerable<T> source)
{
    int hash = 0;
    foreach (T element in source.OrderBy(x => x, Comparer<T>.Default))
    {
        // f is any function/code you like returning int
        hash = f(hash, element);
    }
    return hash;
}

This has some significant benefits in that the combining operations possible in f can have significantly better hashing properties (distribution of bits for example) but this comes at significantly higher cost. The sort is O(n log n) and the required copy of the collection is a memory allocation you can't avoid given the desire to avoid modifying the original. GetHashCode implementations should normally avoid allocations entirely. One possible implementation of f would be similar to that given in the last example under the Addition section (e.g. any constant number of bit shifts left followed by a multiplication by a prime - you could even use successive primes on each iteration at no extra cost, since they only need be generated once).

That said, if you were dealing with cases where you could calculate and cache the hash and amortize the cost over many calls to GetHashCode this approach may yield superior behaviour. Also the latter approach is even more flexible since it can avoid the need to use the GetHashCode on the elements if it knows their type and instead use per byte operations on them to yield even better hash distribution. Such an approach would likely be of use only in cases where the performance was identified as being a significant bottleneck.

Finally, if you want a reasonably comprehensive and fairly non-mathematical overview of the subject of hash codes and their effectiveness in general, these blog posts would be worthwhile reads, in particular the Implementing a simple hashing algorithm (pt II) post.