We study the adaptive modulation (AM) problem in a network-coded two-way relay channel (NC-TWRC), where each of the two users controls its own bit rate in the $m$-ary quadrature amplitude modulation ($m$-QAM) to minimize the transmission error rate and enhance the spectral efficiency. We show that there exists a strategic complementarity, one user tends to transmit while the other decides to do so in order to enhance the overall spectral efficiency, which is beyond the scope of the conventional single-agent AM scheduling method. We propose a two-player game model parameterized by the signal-to-noise ratios (SNRs) of two user-to-user channels and prove that it is a supermodular game where there always exist the extremal pure strategy Nash equilibria (PSNEs), the largest and smallest PSNEs. We show by simulation results that the extremal PSNEs incur a similar bit error rate (BER) as the conventional single-agent AM scheme, but significantly improve the spectral efficiency in the NC-TWRC system. The study also reveals the Pareto order of the extremal PSNEs: The largest and smallest PSNEs are Pareto worst and best PSNEs, respectively. Finally, we derive the sufficient conditions for the extremal PSNEs to be symmetric and monotonic in channel SNRs. We also discuss how to utilize the symmetry and monotonicity to relieve the complexity in the PSNE learning process.