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 anticrossing of spin-split levels depending on subband index and magnetic field direction. Our experimental observations are reconciled in a 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 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 quantum point contact out-of-plane g factor saturates around the predicted 7.2 bulk value.