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Given a manifold (or, more generally, a developable orbifold) $M_0$ and twoclosed Riemannian manifolds $M_1$ and $M_2$ with a finite covering map to$M_0$, we give a spectral characterisation of when they are equivalentRiemannian covers (in particular, isometric), assuming arepresentation-theoretic condition of "homological wideness": if $M$ is acommon 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 homologygroup of $M$ (it holds, for example, when there exists a rational homologyclass on $M$ whose orbit under $G$ consists of $|G|$ linearly independenthomology classes). We prove that, under this condition, Riemannian coveringequivalence is the same as isospectrality of finitely many twisted Laplacianson the manifolds, acting on sections of flat bundles corresponding to specificrepresentations of the fundamental groups of the manifolds involved. Using thesame methods, we provide spectral criteria for weak conjugacy and strongisospectrality. In the negative curvature case, we formulate an analogue of ourresult for the length spectrum. The proofs are inspired by number-theoreticalanalogues. We study examples where the representation theoretic condition doesand does not hold. For example, when $M_1$ and $M_2$ are commensurablenon-arithmetic closed Riemann surfaces of negative Euler characteristic, thereis always such an $M_0$, and the condition of homological wideness alwaysholds.

Twisted isospectrality, homological wideness and isometry