Orthogonal matrix pairs connected by rescaling of rows and columns

Question

For which choices of column-orthogonal matrix $U\in\mathbb{R}^{m\times n}$ and real diagonal matrices $\Lambda_0$ and $\Lambda_1$ (different from the identity) does it hold that $$ O=\Lambda_0U\Lambda_1 $$ is also column-orthogonal? Equivalently, what solutions are there to the above matrix equation where $O$ and $U$ are column-orthogonal.