  We perform a general computation of the off-shell one-loop divergences in
Einstein gravity, in a two-parameter family of path integral measures,
corresponding to different ways of parametrizing the graviton field, and a
two-parameter family of gauges. Trying to reduce the gauge- and
measure-dependence selects certain classes of measures and gauges respectively.
There is a choice of two parameters (corresponding to the exponential
parametrization and the partial gauge condition that the quantum field be
traceless) that automatically eliminates the dependence on the remaining two
parameters and on the cosmological constant. We observe that the divergences
are invariant under a $\mathbf{Z}_2$ "duality" transformation that (in a
particularly important special case) involves the replacement of the densitized
metric by a densitized inverse metric as the fundamental quantum variable. This
singles out a formulation of unimodular gravity as the unique "self-dual"
theory in this class.
