To a Hodge structure of weight with CM by a field we associate Hodge structures of weight for positive and, under certain circumstances, also for negative. We show that these 'half twists' come up naturally in the KugaSatake varieties of weight two Hodge structures with CM by an imaginary quadratic field.
Bert van GEEMEN. "Half twists of Hodge structures of CM-type." J. Math. Soc. Japan 53 (4) 813 - 833, October, 2001. https://doi.org/10.2969/jmsj/05340813