Here's my feeble attempt:
A = A* is the condition for a Hermitian matrix. So I try expanding both sides in terms of their SVD:
A = USV* and A* = (USV*)* = ... = VSU*
So equating I get:
VSU* = USV*
Now maybe through this I can conclude something about how V and U relate to each other but I don't know how...
Thanks for any help.