The presence/absence of a Berry phase depends on the topology of the manifold of dynamical Jahn-Teller potential minima. We describe in detail the relation between these topological properties and the way in which the lowest two adiabatic potential surfaces become locally degenerate. We illustrate our arguments through spherical generalizations of the linear and cases, relevant for the physics of fullerene ions. Our analysis allows us to classify all of the spherical Jahn-Teller systems with respect to the Berry phase. Its absence can, but does not necessarily, lead to a nondegenerate ground state.