  A twist construction for manifolds with torus action is described
generalising certain T-duality examples and constructions in hypercomplex
geometry. It is applied to complex, SKT, hypercomplex and HKT manifolds to
construct compact simply-connected examples. In particular, we find
hypercomplex manifolds that admit no compatible HKT metric, and HKT manifolds
whose Obata connection has holonomy contained in $SL(n,\mathbb H)$.
