How to check if all list items have the same value and return it, or return an “otherValue” if they don’t?

var val = yyy.First().Value;
return yyy.All(x=>x.Value == val) ? val : otherValue; 

Cleanest way I can think of. You can make it a one-liner by inlining val, but First() would be evaluated n times, doubling execution time.

To incorporate the "empty set" behavior specified in the comments, you simply add one more line before the two above:

if(yyy == null || !yyy.Any()) return otherValue;

Good quick test for all equal:

collection.Distinct().Count() == 1

Though you certainly can build such a device out of existing sequence operators, I would in this case be inclined to write this one up as a custom sequence operator. Something like:

// Returns "other" if the list is empty.
// Returns "other" if the list is non-empty and there are two different elements.
// Returns the element of the list if it is non-empty and all elements are the same.
public static int Unanimous(this IEnumerable<int> sequence, int other)
{
    int? first = null;
    foreach(var item in sequence)
    {
        if (first == null)
            first = item;
        else if (first.Value != item)
            return other;
    }
    return first ?? other;
}

That's pretty clear, short, covers all the cases, and does not unnecessarily create extra iterations of the sequence.

Making this into a generic method that works on IEnumerable<T> is left as an exercise. :-)

return collection.All(i => i == collection.First())) 
    ? collection.First() : otherValue;.

Or if you're worried about executing First() for each element (which could be a valid performance concern):

var first = collection.First();
return collection.All(i => i == first) ? first : otherValue;