If a pair of $3\times 3$ rotation matrices $R_A$ and $R_B$ satisfies $$R_B = R_C R_A R_C^T $$ where $R_C$ is an unknown $3\times 3$ rotation matrix. Can $R_C$ be uniquely determined by the congruent relation between $R_A$ and $R_B$?
I guess the answer is yes because we can uniquely determine $R_C$ from the relation $R_B = R_C R_A$ and the congruent relation doesn't introduce more freedoms.