  The symmetries of two-dimensional supersymmetric sigma models on target
spaces with covariantly constant forms associated to special holonomy groups
are analysed. It is shown that each pair of such forms gives rise to a new one,
called a Nijenhuis form, and that there may be further reductions of the
structure group. In many cases of interest there are also covariantly constant
one-forms which also give rise to symmetries. These geometries are of interest
in the context of heterotic supergravity solutions and the associated
reductions are studied from a spacetime point of view via the Killing spinor
equations.
