  We present transport experiments performed in high quality quantum point
contacts embedded in a GaAs two-dimensional hole gas. The strong spin-orbit
interaction results in peculiar transport phenomena, including the previously
observed anisotropic Zeeman splitting and level-dependent effective g-factors.
Here we find additional effects, namely the crossing and the anti-crossing of
spin-split levels depending on subband index and magnetic field direction. Our
experimental observations are reconciled in an heavy hole effective spin-orbit
Hamiltonian where cubic- and quadratic-in-momentum terms appear. The spin-orbit
components, being of great importance for quantum computing applications, are
characterized in terms of magnitude and spin structure. In the light of our
results, we explain the level dependent effective g-factor in an in-plane
field. Through a tilted magnetic field analysis, we show that the QPC
out-of-plane g-factor saturates around the predicted 7.2 bulk value.
