In this paper, the spatial modulation (SM) technique is employed at the source and relay nodes in a full-duplex twoway relay channel (FD-TWRC) to support spectral-efficient bidirectional communications while guaranteeing a low cost implementation. Maximum likelihood (ML) detectors are employed at each node that is subject to an intrinsic self-loop interference (SI). We first propose a tight upper bound on the average bit error probability (ABEP). Then based on the ABEP upper bound, an asymptotic ABEP expression is derived in the high SNR regime. Exploiting the asymptotic ABEP, an exact SNR threshold for the selection between FD-TWRC-SM and half-duplex (HD)-TWRCSM is derived in a closed form, which sheds light on when it is beneficial to select the FD (or HD) mode. In addition, the power allocation (PA) among sources and relay is investigated, through which an optimal PA factor in terms of ABEP is obtained. All analytical results derived in this paper are verified by Monte Carlo simulations, from which some new insights are obtained on the performance of FD-TWRC-SM.