We will discuss some "heat-type" spectral invariants for a pair $(M,f)$ consisting of a compact Riemannian orbifold, $M$, and an isometry, $f$, of $M$, and will show how to obtain inverse results from these invariants involving the "equivariant" spectrum of $M$: the eigenvalues of the Laplace operator plus the representations of the isometry group of $M$ on the eigenspaces. We will also sketch some applications to the theory of toric orbifolds. (This is joint work with Emily Dryden and Rosa Sena-Dias.)