  The existing classification of homogeneous quaternionic spaces is not
complete. We study these spaces in the context of certain $N=2$ supergravity
theories, where dimensional reduction induces a mapping between {\em special}
real, K\"ahler and quaternionic spaces. The geometry of the real spaces is
encoded in cubic polynomials, those of the K\"ahler and quaternionic manifolds
in homogeneous holomorphic functions of second degree. We classify all cubic
polynomials that have an invariance group that acts transitively on the real
manifold. The corresponding K\"ahler and quaternionic manifolds are then
homogeneous. We find that they lead to a well-defined subset of the normal
quaternionic spaces classified by \Al\ (and the corresponding special K\"ahler
spaces given by Cecotti), but there is a new class of rank-3 spaces of
quaternionic dimension larger than 3. We also point out that some of the rank-4
\Al\ spaces were not fully specified and correspond to a finite variety of
inequivalent spaces. A simpler version of the equation that underlies the
classification of this paper also emerges in the context of $W_3$ algebras.
