We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-Hubsch. Their duality matches cohomology classes of two Calabi--Yau orbifolds. When both orbifolds are equipped with an automorphism $s$ of the same order, our mirror duality involves the weight of the action of $s^*$ on cohomology. In particular, it matches the respective $s$-fixed loci, which are not Calabi-Yau in general. When applied to K3 surfaces with non-symplectic automorphism $s$ of odd prime order, this provides a proof that Berglund-Hubsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.
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