Unitary operators with decomposable corners

We study pairs $(U,L_0)$, where $U$ is a unitary operator in $H$ and $L_0\subset H$ is a closed subspace, such that $$P_{L_0}U|_{L_0}:L_0\to L_0$$ has a singular value decomposition. Abstract characterizations of this condition are given, as well as relations to the geometry of projections and pairs of projections. Several concrete examples are examined.