  We study the representation of a finite group acting on the cohomology of a
non-degenerate, invariant hypersurface of a projective toric variety. We deduce
an explicit description of the representation when the toric variety has at
worst quotient singularities. As an application, we conjecture a
representation-theoretic version of Batyrev and Borisov's mirror symmetry
between pairs of Calabi-Yau hypersurfaces, and prove it when the hypersurfaces
are both smooth or have dimension at most 3. An interesting consequence is the
existence of pairs of Calabi-Yau orbifolds whose Hodge diamonds are mirror,
with respect to the usual Hodge structure on singular cohomology.
