  We introduce a type of Riemannian geometry in nine dimensions, which can be
viewed as the counterpart of selfduality in four dimensions. This geometry is
related to a 9-dimensional irreducible representation of ${\bf SO}(3) \times
{\bf SO} (3)$ and it turns out to be defined by a differential 4-form.
Structures admitting a metric connection with totally antisymmetric torsion and
preserving the 4-form are studied in detail, producing locally homogeneous
examples which can be viewed as analogs of self-dual 4-manifolds in dimension
nine.
