We represent and classify pairs of commuting isometries (V₁, V₂) acting on Hilbert spaces such that their cross commutator is compact and normal. 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.

The first named author is supported by the MATRICS research grant, SERB (File Number: MTR/2022/000921). The research of the second and third named authors is supported in part by TARE (TAR/2022/000063) by SERB, Department of Science & Technology (DST), Government of India. The fourth-named author's research was supported by the NBHM Postdoctoral Fellowship (Order No. 0204/10/(24)/2023/R&D-II/2802) during his postdoctoral tenure at IIT Goa.