We identify the twisted sectors of a compact simplicial toric variety. We do the same for a generic nondegenerate Calabi-Yau hypersurface of an $n$-dimensional simplicial Fano toric variety and then explicitly compute $h^{1,1}_{\orb}$ and $h^{n-2,1}_{\orb}$ for the hypersurface. We give applications to the orbifold string theory conjecture and orbifold mirror symmetry.