A space with some elements of quantization/discreteness may have classical continuous symmetries, but only if things are
arranged in an ad hoc manner. Typically quantization/discretization of spacetime observables does lead to departures from
classical spacetime symmetries. So clearly spacetime-symmetry tests should be a core area of quantum-spacetime
phenomenology, but to be pursued with the awareness that spacetime quantization does not automatically affect spacetime
symmetries (it typically does but not automatically).