In this paper, we study operator-theoretic properties of the compressed shift operators 94e87ddb5926491a9114fdcb0ca3" title="Click to view the MathML source">Sz1 and 9bbcc2098e0b7bdddb8881f8863ac4" title="Click to view the MathML source">Sz2 on complements of submodules of the Hardy space over the bidisk H2(D2). Specifically, we study Beurling-type submodules – namely submodules of the form θH2(D2) for θ inner – using properties of Agler decompositions of θ to deduce properties of 94e87ddb5926491a9114fdcb0ca3" title="Click to view the MathML source">Sz1 and 9bbcc2098e0b7bdddb8881f8863ac4" title="Click to view the MathML source">Sz2 on model spaces e5caf58639359f5b5" title="Click to view the MathML source">H2(D2)⊖θH2(D2). Results include characterizations (in terms of θ ) of when a commutator 9b897b7d26488b119cb7464cd52"> has rank n and when subspaces associated to Agler decompositions are reducing for 94e87ddb5926491a9114fdcb0ca3" title="Click to view the MathML source">Sz1 and 9bbcc2098e0b7bdddb8881f8863ac4" title="Click to view the MathML source">Sz2. We include several open questions.