A correspondence between dynamical groups of Schroedinger equations and dynamical groups of Hamilton＇s equations enables one to intrepret both quantal and classical dynamical symmetries as Lie symmetries of total energy surfaces in a phase space. Suitable choices of Cartesian coordinates in the phase space may often be used to express these energy surfaces, whose symmetries are seldom obvious, as group-invariant manifolds with evident geometric symmetries. These invariant manifolds retain their form when a Hamiltonian is subjected to a wide variety of perturbations and may sometimes be constructed without knowing the transformation between their Cartesian coordinates and laboratory coordinates.