This question is continuation of an earlier question asked in Matrices which are both unitary and Hermitian
Consider the unitary group $U(n^2)$ and consider the subset $R$ of Hermitian Unitary matrices. We want to find different conjugacy classes of $R$ by the subgroup action of $U(n)\otimes U(n)$, i.e. $R_1$ ~ $R_2$ iff $\exists U\otimes V$ such that $R_1=U\otimes V R_2 U^*\otimes V^*$. Let $\mathcal{R}$ be the set of such conjugacy classes. My question is, how much is known about thee structure of $\mathcal{R}$, i.e. whether this is finite, or finitely generated or isomorphic to some known object. Also, can someone suggest any paper/article discussing the above problem.