We present time-resolved optical spectroscopy of the counterpart to the high-inclination black hole low-mass X-ray binary Swift J1357.2-0933 in quiescence. Absorption features from the mass donor star were not detected. Instead the spectra display prominent broad double-peaked H alpha emission and weaker He I emission lines. From the H alpha peak-to-peak separation, we constrain the radial velocity semi-amplitude of the donor star to K-2 > 789 km s(-1). Further analysis through radial velocity and equivalent width measurements indicates that the H alpha line is free of variability due to S-wave components or disc eclipses. From our data and previous observations during outburst, we conclude that long-term radial velocity changes ascribed to a precessing disc were of low amplitude or not present. This implies that the centroid position of the line should closely represent the systemic radial velocity, gamma. Using the derived gamma = -150 km s(-1) and the best available limits on the source distance, we infer that the black hole is moving towards the plane in its current Galactic orbit unless the proper motion is substantial. Finally, the depth of the central absorption in the double-peaked profiles adds support for Swift J1357.2-0933 as a high-inclination system. On the other hand, we argue that the low hydrogen column density inferred from X-ray fitting suggests that the system is not seen edge-on.