  Necessary and sufficient conditions for the existence of a
hyper-parahermitian connection with totally skew-symmetric torsion
(HPKT-structure) are presented. It is shown that any HPKT-structure is locally
generated by a real (potential) function. An invariant first order differential
operator is defined on any almost hyper-paracomplex manifold showing that it is
two-step nilpotent exactly when the almost hyper-paracomplex structure is
integrable. A local HPKT-potential is expressed in terms of this operator.
Examples of (locally) invariant HPKT-structures with closed as well as
non-closed torsion 3-form on a class of (locally) homogeneous hyperparacomplex
manifolds (some of them compact) are constructed.
