We report on the stable picosecond and femtosecond pulse generation from the bidirectional erbium-doped all-fiber ring laser hybridly mode-locked with a coaction of a single-walled carbon nanotube-based saturable absorber and nonlinear polarization evolution that was introduced through the insertion of the short-segment polarizing fiber. Depending on the total intracavity dispersion value, the laser emits conservative solitons, transform-limited Gaussian pulses, or highly chirped stretched pulses with almost 20 nm wide parabolic spectrum in both clockwise (CW) and counterclockwise (CCW) directions of the ring. Owing to the polarizing action in the cavity, we have demonstrated for the first time, to the best of our knowledge, an efficient tuning of soliton pulse characteristics for both CW and CCW channels via an appropriate polarization control. We believe that the bidirectional laser presented may be highly promising for gyroscopic and other dual-channel applications.