Twist geometry of the c-map

We discuss the geometry of the c-map from projective special K\"ahler to quaternionic K\"ahler manifolds using the twist construction to provide a global approach to Hitchin's description. As found by Alexandrov et al. and Alekseevsky et al. this is related to the quaternionic flip of Haydys. We prove uniqueness statements for several steps of the construction. In particular, we show that given a hyperK\"ahler manifold with a rotating symmetry, there is essentially only a one parameter degree of freedom in constructing a quaternionic K\"ahler manifold of the same dimension. We demonstrate how examples on group manifolds arise from this picture.