  A Hodge structure V of weight k on which a CM field acts defines, under
certain conditions, a Hodge structure of weight k-1, its half twist. In this
paper we consider hypersurfaces in projective space with a cyclic automorphism
which defines an action of a cyclotomic field on a Hodge substructure in the
cohomology. We determine when the half twist exists and relate it to the
geometry and moduli of the hypersurfaces. We use our results to prove the
existence of a Kuga-Satake correspondance for certain cubic 4-folds.
