In multivalley systems, the valley pseudospin offers rich physics going from encoding of information by its polarization (valleytronics), to exploring novel phases of matter when its degeneracy is changed. Here, by strain engineering, we reveal fully valley-polarized quantum Hall phases in the Pb_{1-x}Sn_{x}Se Dirac system. Remarkably, when the valley energy splitting exceeds the fundamental band gap, we observe a "bipolar quantum Hall phase," heralded by the coexistence of hole and electron chiral edge states at distinct valleys in the same quantum well. This suggests that spatially overlaid counterpropagating chiral edge states emerging at different valleys do not interfere with each other.