How to prove that if $MM^TU=U\Sigma^2$ and $M^TMV=V\Sigma^2$ then $U$ and $V$ are orthogonal?

Question

In the context of Singular Value Decomposition. Suppose $U$ and $V$ are invertible. How to prove that if $$MM^TU=U\Sigma^2$$ and $$M^TMV=V\Sigma^2$$

then $U$ and $V$ are orthogonal?

In the context of singular value decomposition, ($\Sigma$ is diagonal with the eigenvalues of M )