We analyze the operation of a relativistic backward-wave oscillator, in which two waves with opposite directions of rotation of the azimuthal structure of the high-frequency field interact with the electron beam simultaneously. Based on the nonstationary model of such a tube, several characteristic features of dynamics of this system are identified, both in the autonomous regime of its operation, and under the action of an external-signal source. The list of such features includes the instability of azimuthally standing waves, the existence of stable regimes of stationary generation of two waves with different frequencies and opposite rotation directions, and the possibility of spatial self-modulation of the output radiation of a high-frequency oscillator.