Symmetry-enforced topological parity: revisiting $2\mathbb{Z}$ classifications beyond cellwise symmetry actions

Momentum-dependent symmetry actions can turn the even-integer constraint of a $2\mathbb{Z}$ classification of topological insulators and superconductors into an odd-integer constraint without changing the symmetry algebra. For internal and order-two crystalline symmetries in up to four spatial dimensions, we derive parity relations between the Chern or winding number of a gapped Hamiltonian and topological invariants of its momentum-dependent antiunitary symmetry matrices. Whenever the symmetry action enforces odd parity, any symmetry-preserving gapped phase must be topologically nontrivial. Using cellwise actions, which are represented by momentum-independent matrices in a fixed unit-cell basis, as a reference, we determine how locality constrains the realization of odd topological numbers. Explicit model constructions and no-go theorems distinguish three cases: odd values can be realized by finite-range symmetry actions, can be realized by exponentially decaying quasilocal actions but not by finite-range actions, or are impossible in finite-dimensional Bloch systems.