Current-induced spin-orbit torques (SOTs) in structurally asymmetric multilayers have been used to efficiently manipulate magnetization. In a structure with vertical symmetry breaking, a damping-like SOT can deterministically switch a perpendicular magnet, provided an in-plane magnetic field is applied. Recently, it has been further demonstrated that the in-plane magnetic field can be eliminated by introducing a new type of perpendicular field-like SOT via incorporating a lateral structural asymmetry into the device. Typically, however, when a current is applied to such devices with combined vertical and lateral asymmetries, both the perpendicular field-like torque and the damping-like torque coexist, hence jointly affecting the magnetization switching behavior. Here, we study perpendicular magnetization switching driven by the combination of the perpendicular field-like and the damping-like SOTs, which exhibits deterministic switching mediated through domain wall propagation. It is demonstrated that the role of the damping-like SOT in the deterministic switching is highly dependent on the magnetization direction in the domain wall. By contrast, the perpendicular field-like SOT is solely determined by the relative orientation between the lateral structural asymmetry and the current direction, regardless of the magnetization direction in the domain wall. The experimental results further the understanding of SOTs-induced switching, with implications for spintronic devices.