  Motivated by generalized geometry (\`a la Hitchin), we discuss the
integrability conditions for four natural almost complex structures on the
product bundle ${\mathcal Z}\times {\mathcal Z}\to M$, where ${\mathcal Z}$ is
the twistor space of a Riemannian 4-manifold $M$ endowed with a metric
connection $D$ with skew-symmetric torsion. These structures are defined by
means of the connection $D$ and four (K\"ahler) complex structures on the
fibres of this bundle. Their integrability conditions are interpreted in terms
of Weyl geometry and this is used to supply examples satisfying the conditions.
