  In the present paper, we study compact complex manifolds admitting a
Hermitian metric which is SKT and CYT and whose Bismut torsion is parallel. We
first obtain a characterization of the universal cover of such manifolds as a
product of a Kaehler Ricci-flat manifold with a Bismut flat one. Then, using a
mapping torus construction, we provide non-Bismut flat examples. The existence
of generalized Kaehler structures is also investigated.
