Is there a list of 'natural' complexity classes closed under complement? Some that I could think of are P, ZPP, BPP, NP $\cap$ co-NP, PH, PP and PSPACE but surely there are others. Wikipedia and the Complexity-Zoo don't have such a list as far as I can tell but perhaps there are other resources?
All deterministic classes are (almost trivially) closed under complement:
Given a deterministic TM $M$ for a language $L$, one obtains a deterministic TM for its complement by simply running $M$ and then "flipping" the answer. The machine runs in the same time and space asymptotically. By applying the same proof to the complement class, it follows that $L=co-L$.
The same obviously holds for all classes where the result can be safely "flipped", e.g. $ZPP$. Nondeterministic classes obviously fail this requirement.
Nonderministic Logspace (NL) is closed under complement. That is, $NL=coNL$. This non-trivial result is known as (or rather, is the consequence of) the Immerman–Szelepcsényi theorem.
Shaull already mentioned NL. The same thing also applies to all super-logarithmic space classes. The only other examples I can find for which I don't know of as-natural
equivalent definitions from which closedness under complement trivially follows are:
SAC and each of its positive levels