  The twist construction is a geometric T-duality that produces new manifolds
from old, works well with for example hypercomplex structures and is easily
inverted. It tends to destroy properties such as the hyperK\"ahler condition.
On the other hand modifications preserve the hyperK\"ahler property, but do not
have an obvious inversion. In this paper we show how elementary deformations
provide a link between the two constructions, and use the twist construction to
build hyperK\"ahler and strong HKT structures. In the process, we provide a
full classification of complete hyperK\"ahler four-manifolds with
tri-Hamiltonian symmetry and study a number singular phenomena in detail.
