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 spinsplit 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-inmomentum 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.Spin-orbit interaction (SOI) is a relativistic effect that couples the motion of an electron to its spin [1]. For two-dimensional electron gases in the conduction band of III-V materials, SOI originates from bulk inversion asymmetry (Dresselhaus SOI [2]) and structure inversion asymmetry (Rashba SOI [3]) and takes the form, with σ the Pauli matrices and k the in-plane wavevector [4]. For two-dimensional hole gases (2DHGs) in the valence band of GaAs the situation is very different. Because of the non-zero orbital angular momentum, bulk SOI, and confinement in growth direction, SOI for holes is expected to be more pronounced than for their electronic counterparts, mainly of Rashba type and cubic in k [5,6]. The relevance of an additional term, quadratic in k and proportional to the in-plane components of the applied magnetic field B, was recently proposed [7][8][9][10]. Such a term is unique for heavy holes and very useful for exploiting SOI for quantum computing applications [7]. In this manuscript we show how the cubic and quadratic terms present in the bulk Hamiltonian can be separately addressed in the magnetoconductance of a quantum point contact (QPC) embedded in a 2DHG. Furthermore, our results offer a better understanding of the physics of ptype QPCs in terms of level dependent in-plane and outof-plane g-factors (g and g ⊥ respectively) and allow us to measure the bulk g ⊥ . The latter is particularly interesting, since the bulk g-factor anisotropy of p-type GaAs [11-13] makes it impossible to directly measure g ⊥ with conventional transport techniques [14]. Theoretical predictions for a [001]-growth 2DHG estimate g = 0 and g ⊥ = 7.2 [5,15]. It was argued [16] that in a QPC, in the limit of high subband index n, g ⊥ should approach the bulk value. So far, despite the tendency of g ⊥ to increase with n, this prediction was not experimentally confirmed.The experiment was performed using a carbon doped GaAs 2DHG grown along the [001] direction. A strong Rashba SOI is expected here due to the asymmetry of the confinement potential. A complete ...