We consider relaying strategies for memoryless two-way relay channels, where the signal transmitted by a relay depends only on its previously received signal. For binary antipodal signaling under the general case of different SNRs on the terminal-relay channels, we consider two classes of relay strategies: abs-based strategies in which the processing at the relay is solely based on the absolute value of the received signal, and non-abs-based strategies. We analyze and optimize the performance of existing and new schemes for two-way relay channels under an average power constraint, including abs-based and non-abs-based versions of amplify and forward (AF), decode and forward (DF), and estimate and forward (EF). Additionally, we optimize the relay function via functional analysis such that the average probability of error is minimized in the high signal-to-noise ratio (SNR) regime. Finally, we show that all these results can also be generalized to higher order constellations.