We report quantitative experimental and theoretical results revealing the tunability of spin splitting in high-mobility two-dimensional GaAs hole systems, confined to either a square or a triangular quantum well, via the application of a surface-gate bias. The spin splitting depends on both the hole density and the symmetry of the confinement potential and is largest for the highest densities in asymmetric potentials. In the triangular well, when the spin splitting is sufficiently large, our measured commensurability oscillations, induced by a one-dimensional periodic potential, exhibit two frequencies providing clear evidence for spin-resolved ballistic transport.