A bstract We study four-dimensional gauge theories with arbitrary simple gauge group with 1-form global center symmetry and 0-form parity or discrete chiral symmetry. We canonically quantize on 𝕋 3 , in a fixed background field gauging the 1-form symmetry. We show that the mixed 0-form/1-form ’t Hooft anomaly results in a central extension of the global-symmetry operator algebra. We determine this algebra in each case and show that the anomaly implies degeneracies in the spectrum of the Hamiltonian at any finite- size torus. We discuss the consistency of these constraints with both older and recent semiclassical calculations in SU( N ) theories, with or without adjoint fermions, as well as with their conjectured infrared phases.