  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.
