  We consider the $G$-invariant spectrum of the Laplacian on an orbit space
$M/G$ where $M$ is a compact Riemannian manifold and $G$ acts by isometries. We
generalize the Sunada-Pesce-Sutton technique to the $G$-invariant setting to
produce pairs of isospectral non-isometric orbit spaces. One of these spaces is
isometric to an orbifold with constant sectional curvature whereas the other
admits non-orbifold singularities and therefore has unbounded sectional
curvature. We therefore show that constant sectional curvature and the presence
of non-orbifold singularities are inaudible to the G-invariant spectrum.
