Let $T \in \mathbb{R}^{n \times n}$ be a product of two symmetric matrices $A, B \in \mathbb{R}^{n \times n}$:
$$T = AB$$
By the Spectral Theorem, $A$ and $B$ each have an eigendecomposition $A = UD_AU^T$ and $B = VD_BV^T$, where $U,V \in \mathbb{R}^{n \times n}$ are orthogonal matrices and $D_A, D_B \in \mathbb{R}^{n \times n}$ are diagonal matrices:
$$T = UD_AU^TVD_BV^T$$
Suppose both $D_A$ and $D_B$ have at least one non-zero diagonal entry. If I know what $U$ is, is it possible to determine what $V$ is?