Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors? Given the product $M$ of a unitary matrix $U$ (i.e. $U^\dagger U=1$) and an orthogonal matrix $O$ (i.e. $O^TO=1$), can it be easily inverted without knowing $U$ and $O$?
Sure enough, if $M=UO$, then $M^{-1}=O^TU^\dagger$. But assuming you only know that $M$ is composed in such a way, but not how $U$ and $O$ actually look, does there still exist a simple formula for $M^{-1}$?

Another direction, conceptually the same:
$$MM^T = U\overbrace{OO^T}^IU^T = UU^T$$
$$\Rightarrow (MM^T)^{-1} = U^*U^\dagger = (MM^T)^* = M^*M^\dagger$$
$$\Rightarrow MM^TM^*M^\dagger=I$$
