  The purpose of this paper is to review some combinatorial ideas behind the
mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in
Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the
notion of a Gorenstein polytope of index r. A natural combinatorial duality for
d-dimensional Gorenstein polytopes of index r extends the well-known polar
duality for reflexive polytopes (case r=1). We consider the Borisov duality
between two nef-partitions as a duality between two Gorenstein polytopes P and
P^* of index r together with selected special (r-1)-dimensional simplices S in
P and S' in P^*. Different choices of these simplices suggest an interesting
relation to Homological Mirror Symmetry.
