Suppose you have a unitary matrix $U$ such that $\overline{U}U=D$ for some diagonal unitary matrix $D$ (everything is taking place over $\mathbb{C}$). This is equivalent to a couple of other conditions:

1) $DU=U^T$

2) $DUD=U$

Furthermore, this implies that $U^2$ is symmetric. So most unitary matrices do not satisfy this property. But when they do, is it true that $D^2=I$, which is equivalent to saying that $D$ commutes with $U$?