Has this been done in other tests already? If I'd seen it, I'd say why are you adding the &.
I would make IterSwapT an alias for decltype(std::ranges::iter_swap), and then I would check both std::invocable_v<IterSwapT, blah> and a few std::invocable_v<IterSwapT&, blah> and std::invocable_v<IterSwapT&&, blah>. (Remember reference-collapsing means that with your current definition, all three of IterSwapT{,&,&&} are equivalent to IterSwapT&, and that strikes me as less-useful-than-it-could-be.)

Based on http://eel.is/c++draft/customization.point.object#2, I would just do std::semiregular<std::remove_cv_t<IterSwapT>>.

nit: explicit plz

https://quuxplusone.github.io/blog/2021/04/03/static-constexpr-whittling-knife/#friendness-to-befriend-an-entity :)

Here and line 54: I'm worried that your implementation of this function is unnecessary and/or indistinguishable from the default ranges::swap, which means you're not actually testing that it's called. How about replacing these implementations with

friend constexpr void iter_swap(HasIterSwap& a, HasIterSwap& b) {
    called1 = true;
}
friend constexpr void iter_swap(HasIterSwap& a, bool& b) {
    called2 = true;
}

and then writing tests that assert(called1 && !called2) (or whatever you expect to happen).
For constexpr-friendliness, you'll probably need to make your constructor explicit HasIterSwap(bool& c1, bool& c2) : called1(c1), called2(c2) {}. Slack me if this doesn't make sense.

Yes — ranges::iter_swap(++i, ++j) is supposed to increment i only once, not twice.
But this is physically obvious. The Standard needs to mention it because the Standard is pretending that these are expressions-not-function-calls. We don't have to test it; you can remove this TODO.

Minor thing: I like to split test different test cases by enclosing them in a different scope. It makes it easier to see that they are different test cases and that they don't share anything between themselves. So I'd do:

{
    int value1 = 0;
    int value2 = 0;
    HasIterSwap a(value1), b(value2);
    std::ranges::iter_swap(a, b);
    assert(value1 == 1 && value2 == 1);
}
{
    int value1 = 0;
    int value2 = 0;
    HasRangesSwap c(value1), d(value2);
    HasRangesSwapWrapper cWrapper(c), dWrapper(d);
    std::ranges::iter_swap(cWrapper, dWrapper);
    assert(value1 == 1 && value2 == 1);
}

// etc...

This is the only place you test std::ranges::iter_swap(lvalue, lvalue). Please expand the tests below to test them also with lvalues of type HasRangesSwapWrapper, A*, etc., at least enough to hit each different overload of iter_swap once with at least one lvalue.

You never test std::ranges::iter_swap(lvalue, rvalue) or std::ranges::iter_swap(rvalue, lvalue), but I don't think those need testing, as long as you hit the (lvalue, lvalue) cases.

It seems like this is testing the order in which std::swap performs its moves — is that right? If so, I don't think it belongs in libcxx/test/std/ at all. Ditto throughout.