It is shown that the Hilbert space decomposition of a bifinite correspondence between II$_1$ factors (in the sense of Connes) is the same as the purely algebraic decomposition of its bounded vectors. This makes natural the systematic study of pairs of finite index subfactors, for which a combinatorial and a spectral invariant are defined by analogy with the invariants of a pair of closed subspaces of a Hilbert space. Some simple examples are calculated.