By comparison with other systems, vibronic coupling between an h-type vibration and an H-type electronic level in a Hxh Jahn-Teller system would be expected to result in a vibronic H-type ground state. However, it is already known that an A state can cross over the H state at a given coupling strength and become the ground state if the ground adiabatic potential energy surface contains minima of D-3d symmetry. This is an unusual property of the Hxh Jahn-Teller system. In this article, the physics behind the crossover is analyzed in terms of competition between tunneling paths between D-3d wells of C-1 and C-2 symmetries. The Hxh Jahn-Teller system is relevant to some fullerenes and other icosahedral complexes.