  We set up a consistent background field formalism for studying the
renormalization group (RG) flow of gravity coupled to $N_f$ Dirac fermions on
maximally symmetric backgrounds. Based on Wetterich's equation we perform a
detailed study of the resulting fixed point structure in a projection including
the Einstein-Hilbert action, the fermion anomalous dimension, and a specific
coupling of the fermion bilinears to the spacetime curvature. The latter
constitutes a mass-type term which breaks chiral symmetry explicitly. Our
analysis identifies two infinite families of interacting RG fixed points which
are viable candidates to provide a high-energy completion through the
asymptotic safety mechanism. The fixed points exist for all values of $N_f$
outside of a small window situated at low values $N_f$ and become weakly
coupled in the large $N_f$-limit. Symmetry-wise, they correspond to
"quasi-chiral" and "non-chiral" fixed points. The former come with enhanced
predictive power, fixing one of the couplings via the asymptotic safety
condition. Moreover, the interplay of the fixed points allows for cross-overs
from the non-chiral to the chiral fixed point, giving a dynamical mechanism for
restoring the symmetry approximately at intermediate scales. Our discussion of
chiral symmetry breaking effects provides strong indications that the topology
of spacetime plays a crucial role when analyzing whether quantum gravity admits
light chiral fermions.
