  The most fundamental example of mirror symmetry compares the Fermat
hypersurfaces in P^n and P^n/G, where G is a finite group that acts on P^n and
preserves the Fermat hypersurface. We generalise this to hypersurfaces in
Grassmannians, where the picture is richer and more complex. There is a finite
group G that acts on the Grassmannian Gr(n,r) and preserves an appropriate
Calabi-Yau hypersurface. We establish how mirror symmetry, toric degenerations,
blow-ups and variation of GIT relate the Calabi-Yau hypersurfaces inside
Gr(n,r) and Gr(n,r)/G. This allows us to describe a compactification of the
Eguchi-Hori-Xiong mirror to the Grassmannian, inside a blow-up of the quotient
of the Grassmannian by G.
