  We illustrate a general reconstruction procedure for mimetic gravity.
Focusing on a bouncing cosmological background, we derive general properties
that must be satisfied by the function $f(\Box\phi)$ implementing the limiting
curvature hypothesis. We show how relevant physical information can be
extracted from power law expansions of $f$ in different regimes, corresponding
e.g. to the very early universe or to late times. Our results are then applied
to two specific models reproducing the cosmological background dynamics
obtained in group field theory and in loop quantum cosmology, and we discuss
the possibility of using this framework as providing an effective field theory
description of quantum gravity. We study the evolution of anisotropies near the
bounce, and discuss instabilities of scalar perturbations. Furthermore, we
discuss two equivalent formulations of mimetic gravity: one in terms of an
effective fluid with exotic properties, the other featuring two distinct
time-varying gravitational "constants" in the cosmological equations.
