The electrical control of perpendicular magnetization without the need for external magnetic fields holds significant potential for next-generation spintronic devices. In this Letter, we have identified a 3m-symmetry dependent field-free switching phenomenon in (111)-oriented Tm_{3}Fe_{5}O_{12} single-crystal films capped with a platinum (Pt) layer. We demonstrate that this distinctive property arises due to the spontaneous breaking of mirror symmetry in magnetocrystalline anisotropy (MCA) for (111)-oriented magnetic films with a cubic structure, which results in a local out-of-plane MCA effective field with a 3m-symmetry dependence on the azimuth angle when the magnetization lies in the (111) plane. The observed field-free switching, facilitated by this MCA effective field, is validated by numerical simulations based both on macro-spin model and on micromagnetic theory. Our findings not only underscore the instrumental role of intrinsic MCA in enabling field-free magnetization switching but also enrich the fundamental understanding of the underlying switching dynamics.