  We study the perturbative renormalization of quantum gauge theories in the
Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (2007)
that the quantum counterparts of gauge symmetries -- the so-called
Ward--Takahashi and Slavnov--Taylor identities -- correspond to Hopf ideals in
the respective renormalization Hopf algebra. We generalize this correspondence
to super- and non-renormalizable Quantum Field Theories, extend it to theories
with multiple coupling constants and add a discussion on transversality. In
particular, this allows us to apply our results to (effective) Quantum General
Relativity, possibly coupled to matter from the Standard Model, as was
suggested by Kreimer (2008). To this end, we introduce different gradings on
the renormalization Hopf algebra and study combinatorial properties of the
superficial degree of divergence. Then we generalize known coproduct and
antipode identities to the super- and non-renormalizable cases and to theories
with multiple vertex residues. Building upon our main result, we provide
criteria for the compatibility of these Hopf ideals with the corresponding
renormalized Feynman rules. A direct consequence of our findings is the
well-definedness of the Corolla polynomial for Quantum Yang--Mills theory
without reference to a particular renormalization scheme.
