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It has been a long-standing open problem to construct a general framework forrelating the spectra of dual theories to each other. Here, we solve thisproblem for the case of one-dimensional quantum lattice models withsymmetry-twisted boundary conditions. In ref. [PRX Quantum 4, 020357],dualities are defined between (categorically) symmetric models that only differin a choice of module category. Using matrix product operators, we constructfrom the data of module functors explicit symmetry operators preservingboundary conditions as well as intertwiners mapping topological sectors of dualmodels onto one another. We illustrate our construction with a family ofexamples that are in the duality class of the spin-$\frac{1}{2}$ Heisenberg XXZmodel. One model has symmetry operators forming the fusion category$\mathsf{Rep}(\mathcal S_3)$ of representations of the group $\mathcal S_3$. Wefind that the mapping between its topological sectors and those of the XXZmodel is associated with the non-trivial braided auto-equivalence of theDrinfel'd center of $\mathsf{Rep}(\mathcal S_3)$.

Dualities in one-dimensional quantum lattice models: topological sectors