Polarization is a crucial component in cell differentiation, development, and motility, but its details are not yet well understood. At the onset of cell locomotion, cells break symmetry to form well-defined cell fronts and rears. This polarity establishment varies across cell types: in Dictyostelium discoideum cells, it is mediated by biochemical signaling pathways and can function in the absence of a cytoskeleton, while in keratocytes, it is tightly connected to cytoskeletal dynamics and mechanics. Theoretical models that have been developed to understand the onset of polarization have explored either signaling or mechanical pathways, yet few have explored mechanochemical mechanisms. However, many motile cells rely on both signaling modules and actin cytoskeleton to break symmetry and achieve a stable polarized state. We propose a general mechanochemical polarization model based on coupling between a stochastic model for the segregation of signaling molecules and a simplified mechanical model for actin cytoskeleton network competition. We find that local linear coupling between minimally nonlinear signaling and cytoskeletal systems, separately not supporting stable polarization, yields a robustly polarized cell state. The model captures the essence of spontaneous polarization of neutrophils, which has been proposed to emerge due to the competition between frontness and backness pathways.