Symmetries are important for understanding equilibrium as well as nonequilibrium properties like transport. In translationally invariant extended systems one might expect symmetry generators to also be homogeneous. Studying qubit circuits with nearest-neighbor U(1) gates we show that this needs not be the case. We find new inhomogeneous screw SU(2) and \({{{{\rm{U}}}}}_{q}({{\mathfrak{sl}}}_{2})\) symmetries whose generators exhibit a spatial quasi-momentum modulation. They can be viewed as a parameter-dependent generalization of the standard rotational symmetry of the Heisenberg model and can be identified by the Ruelle-Pollicott spectrum of a momentum-resolved propagator. Rich integrability structure is reflected also in transport: picking an arbitrary U(1) gate and varying the gate duration one will transition through different phases, including fractal ballistic transport, Kardar-Parisi-Zhang superdiffusion at the critical manifold that also contains helix states, diffusion, and localization. To correctly explain transport the non-local SU(2) symmetries do not matter, while the inhomogeneous local ones that almost commute with the propagator do.
