  Given a manifold (or, more generally, a developable orbifold) $M_0$ and two
closed Riemannian manifolds $M_1$ and $M_2$ with a finite covering map to
$M_0$, we give a spectral characterisation of when they are equivalent
Riemannian covers (in particular, isometric), assuming a
representation-theoretic condition of "homological wideness": if $M$ is a
common finite cover of $M_1$ and $M_2$ and $G$ is the covering group of $M$
over $M_0$, the condition involves the action of $G$ on the first homology
group of $M$ (it holds, for example, when there exists a rational homology
class on $M$ whose orbit under $G$ consists of $|G|$ linearly independent
homology classes). We prove that, under this condition, Riemannian covering
equivalence is the same as isospectrality of finitely many twisted Laplacians
on the manifolds, acting on sections of flat bundles corresponding to specific
representations of the fundamental groups of the manifolds involved. Using the
same methods, we provide spectral criteria for weak conjugacy and strong
isospectrality. In the negative curvature case, we formulate an analogue of our
result for the length spectrum. The proofs are inspired by number-theoretical
analogues. We study examples where the representation theoretic condition does
and does not hold. For example, when $M_1$ and $M_2$ are commensurable
non-arithmetic closed Riemann surfaces of negative Euler characteristic, there
is always such an $M_0$, and the condition of homological wideness always
holds.
