  We develop tools that allow the systematic enumeration of inequivalent
holomorphic orientifolds of Calabi-Yau hypersurfaces in toric fourfolds of
arbitrary Hodge numbers. As examples, we construct an orientifold of the
Calabi-Yau hypersurface with largest known Hodge number $h^{1,1}=491$, as well
as an orientifold of a Calabi-Yau hypersurface with $h^{1,1}=243$ that yields a
large orientifold-odd Hodge number $h^{1,1}_-=120$.
