  We provide a sufficient condition for a general hypersurface in a $\mathbb
Q$-Fano toric variety to be a Calabi-Yau variety in terms of its Newton
polytope. Moreover, we define a generalization of the Berglund-H\"ubsch-Krawitz
construction in case the ambient is a $\mathbb Q$-Fano toric variety with
torsion free class group and the defining polynomial is not necessarily of
Delsarte type. Finally, we introduce a duality between families of Calabi-Yau
hypersurfaces which includes both Batyrev and Berglund-H\"ubsch-Krawitz mirror
constructions. This is given in terms of a polar duality between pairs of
polytopes $\Delta_1\subseteq \Delta_2$, where $\Delta_1$ and $\Delta_2^*$ are
canonical.
