  Given a conformally variational scalar Riemannian invariant $I$, we identify
a sufficient condition for a closed Riemannian manifold to admit finite regular
coverings with many nonhomothetic conformal rescalings with $I$ constant. We
also identify a sufficient condition for the universal cover to admit
infinitely many geometrically distinct periodic conformal rescalings with $I$
constant. Using these conditions, we improve known nonuniqueness results for
the $Q$-curvatures of orders two, four, and six, and establish nonuniqueness
results for higher-order $Q$-curvatures and renormalized volume coefficients.
