Most "normal" non-uniform circuit classes are closed under complement. Just add a negation gate to the output of a circuit, and if necessary, apply De-Morgan's law.
Now there are some natural non-uniform classes of circuits, such as skew circuits of polylog depth which are still closed under complement but under much less obvious proofs.
- Are there natural non-uniform circuit classes that are provably not closed under complement?
- If the above question is too hard, are there natural circuit classes inside P/poly which imply something unexpected if they happen to be closed under complement?
I expect that a class satisfying any of the above will be kind of weird, since if $C_1 \subseteq X \subseteq C_2$ are complexity clases and $X$ is not closed under complement then $C_1\neq C_2$. Since such separations are lacking, by "natural" I mean some class that has been studied before and not made up exclusively to answer this question. On the other hand, I think that it would be interesting to have ways of engineering complexity classes that are not closed under complement.