What is the correct way to compare char ignoring case?

It depends on what you mean by "work for all cultures". Would you want "i" and "I" to be equal even in Turkey?

You could use:

bool equal = char.ToUpperInvariant(x) == char.ToUpperInvariant(y);

... but I'm not sure whether that "works" according to all cultures by your understanding of "works".

Of course you could convert both characters to strings and then perform whatever comparison you want on the strings. Somewhat less efficient, but it does give you all the range of comparisons available in the framework:

bool equal = x.ToString().Equals(y.ToString(), 
                                 StringComparison.InvariantCultureIgnoreCase);

For surrogate pairs, a Comparer<char> isn't going to be feasible anyway, because you don't have a single char. You could create a Comparer<int> though.

As I understand it, there isn't really a way that will "work for all cultures". Either you want to compare characters for some kind of internal, non-displayed-to-the-user reason (in which case you should use the InvariantCulture), or you want to use the CurrentCulture of the user. Obviously, using the user's current culture will mean that you will get different results in different locales, but they will be consistent with what your users in those locales will expect.

Without knowing more about WHY you are comparing two characters, I can't really advise you on which one you should be using.

I would recommend comparing uppercase, and if they don't match then comparing lowercase, just in case the locale's uppercasing and lowercasing logic behave slightly different.

Addendum

For example,

int CompareChar(char c1, char c2)
{
    int  dif;

    dif = char.ToUpper(c1) - char.ToUpper(c2);
    if (diff != 0)
        dif = char.ToLower(c1) - char.ToLower(c2);
    return dif;
}