# When is the transpose of a square unitary matrix also unitary?

If I have a unitary square matrix $U$ ie. $U^{\dagger}U=I$ ( $^\dagger$ stands for complex conjugate and transpose ), then for what cases is $U^{T}$ also unitary. One simple case I can think of is $U=U^{T}$ ( all entries of $U$ are real, where $^T$ stands for transpose ). Are there any other cases ?

In particular, if $U$ is unitary, then $$(U^T)^\dagger U^T = [UU^\dagger]^T = I^T = I$$
$$(U^T)^\dagger = \bar U = (U^\dagger)^T,$$ where $\bar U$ is the complex conjugate of $U$.
Moroever $$(U^T)^\dagger U^T = (U^\dagger)^T U^T = UU^\dagger = I.$$