We study the damping-like spin–orbit torque (DL-SOT) efficiencies in W/Pt/Co/Pt multilayer structures by the current-induced hysteresis loop shift measurement and current-induced magnetization switching measurement. It is known that transition metals W and Pt possess spin Hall ratios with opposite signs, and therefore, the DL-SOT efficiencies in these multilayer structures may become zero with a certain W/Pt thickness combination. In this work, we show that indeed the zero DL-SOT efficiency can be achieved in such a structure, and the efficiency can evolve from negative (W-dominated) to positive (Pt-dominated) depending on the relative thickness of W and Pt. More importantly, we did not observe field-free switching when the W/Pt combination gives zero DL-SOT efficiency, which is in contrast to a recent report [Ma et al., Phys. Rev. Lett. 120, 117703 (2018)]. By further considering a simple spin diffusion model, we find that DL-SOT efficiencies ξDLPt=0.12 and ξDLW=−0.13 for the Pt and W layer, respectively, in our multilayer system. We also show that the Pt(2)/Co(0.5)/Pt(2) symmetric structure is a robust perpendicular magnetization anisotropy multilayer that can be employed on W or other spin Hall materials to characterize their DL-SOT efficiencies.