In 1992, Fannes, Nachtergaele, and Werner classified translation invariant states on quantum spin chains and discovered that they admit a matrix product structure. Such matrix product states are simultaneously good approximations for general states, and natural candidates for ground states of specific local Hamiltonians. Following the observation by Vidal (2004) that matrix product states are ‘‘efficient,’’ the theory took root and is now an indispensable tool in many-body physics and quantum simulation. Recent work in this direction by Movassagh–Schenker (2022) and Nelson–R. (2024) adapted this structure to states generated by disordered matrix products. All such disordered matrix product states are translation co-variant. However both works above only had a ‘‘one-way’’ construction, not a classification. In this talk, I’ll report on some work in progress with Jeffrey Schenker where we successfully classify the translation co-variant states when the underlying probability space is a compact Hausdorff space.