For an $n \times n$ matrix whose SVD is $A = U D V^\top$, what do we know about $UV$?

Question

I'm using a svd (singular value decomposition) function in a programming library that I didn't write. Given a square real nxn matrix, svd returns three values U,S,V where S is a vector designating a diagonal matrix, and U and V are both nxn matrices.

I was expecting to find that $U\times V=I$, but I find that is not the case.

What do we know about $U\times V$ in the singular value decomposition?

What is the name of a decomposition I can search for which gives me $XDX^{-1}$ ?