  We introduce and study a notion of invariant intrinsic torsion geometry which
appears, for instance, in connection with the Bryant-Salamon metric on the
spinor bundle over S^3. This space is foliated by six-dimensional
hypersurfaces, each of which carries a particular type of SO(3)-structure; the
intrinsic torsion is invariant under SO(3). The last condition is sufficient to
imply local homogeneity of such geometries, and this allows us to give a
classification. We close the circle by showing that the Bryant-Salamon metric
is the unique complete metric with holonomy G_2 that arises from
SO(3)-structures with invariant intrinsic torsion.
