In this paper, we maximize the energy efficiency (EE) of full-duplex (FD) two-way relay (TWR) systems under non-ideal power amplifiers (PAs) and non-negligible transmission-dependent circuit power. We start with the case where only the relay operates full duplex and two timeslots are required for TWR. Then, we extend to the advanced case, where the relay and the two nodes all operate full duplex, and accomplish TWR in a single timeslot. In both cases, we establish the intrinsic connections between the optimal transmit powers and durations, based on which the original non-convex EE maximization can be convexified and optimally solved. Simulations show the superiority of FD-TWR in terms of EE, especially when traffic demand is high. The simulations also reveal that the maximum EE of FD-TWR is more sensitive to the PA efficiency, than it is to self-cancellation. The full FD design of FD-TWR is susceptible to traffic imbalance, while the design with only the relay operating in the FD mode exhibits strong tolerance.