In ZFC set theory (or better in NBG set theory, where the language is more flexible with proper classes), we have that every unbounded class of ordinal numbers is a proper subclass of the class On of all ordinals and that every such proper class has a unique bijection (by the enumeration function) with the proper class On.

More generally, every proper class that is (class) well-orderable (that is, that is well-orderable, with every descending section being a set) is also bijective with On. So that, every proper class is bijective with On iff V (the class of all sets) is (class) well-orderable (this is known as the global choice axiom).

So, if we do not have the global choice axiom there is no bijection between On and V, and we are left with at least two bijective-equivalent collections of proper classes.

Question 1: Does ZFC (or NBG) prove more that that concerning the bijective-equivalent collections of proper classes?

Question 2: If we pass to a set theory whose language allows for quantification over proper classes (like KM, the Kelley-Morse set theory), do we know more about these bijective-equivalent collections of proper classes?