It is widely observed that in practical wireless cooperative communication systems, different links may experience different fading characteristics. In this paper, we investigate into the outage probability and channel capacity of two-way amplify-and-forward (TWAF) relaying systems operating over a mixed asymmetric Rician and Rayleigh fading scenario, with different amplification policies (AP) adopted at the relay, respectively. As TWAF relay network carries concurrent traffics towards two opposite directions, both end-to-end and overall performance metrics were considered. In detail, both uniform exact expressions and simplified asymptotic expressions for the end-to-end outage probability (OP) were presented, based on which the system overall OP was studied under the condition of the two source nodes having non-identical traffic requirements. Furthermore, exact expressions for tight lower bounds as well as high SNR approximations of channel capacity of the considered scenario were presented. For both OP and channel capacity, with different APs, effective power allocation (PA) schemes under different constraints were given to optimize the system performance. Extensive simulations were carried out to verify the analytical results and to demonstrate the impact of channel asymmetry on the system performance.