  We represent and classify pairs of commuting isometries $(V_1, V_2)$ acting
on Hilbert spaces that satisfy the condition \[ [V_1^*, V_2] = \text{compact} +
\text{normal}, \] where $[V_1^*, V_2] := V_1^* V_2 - V_2 V_1^*$ is the
cross-commutator of $(V_1, V_2)$. The precise description of such pairs also
gives a complete and concrete set of unitary invariants. The basic building
blocks of representations of such pairs consist of four distinguished pairs of
commuting isometries. One of them relies on some peculiar examples of invariant
subspaces tracing back to Rudin's intricate constructions of analytic functions
on the bidisc. Along the way, we present a rank formula for a general pair of
commuting isometries that looks to be the first of its kind.
