Classical superstring vacua corresponding to c = 9, N = (2,2) superconformal theories lead to Kahler or quaternionic nonlinear sigma models that are characterized in terms of a holomorphic homogeneous function of second degree. The Kahler manifolds may be invariant under the generalized duality invariances known from supergravity. Here we study the symmetry structure of the corresponding quaternionic manifolds, which have a symmetry group that is considerably larger. It turns out that the symmetry structure is encoded in a single function invariant under duality transformations. We discuss a number of examples and exhibit the existence of new symmetries.