The cubic perovskite SrFeO${}_3$ is a prototypical centrosymmetric itinerant magnet that hosts a quadruple-$\yen mathbfQ$ hedgehog–antihedgehog lattice and exhibits a complex magnetic-field–temperature phase diagram. Yet, the microscopic mechanism underlying the emergence of its versatile multiple-$\yen mathbfQ$ phases remains unresolved. Here, we reveal the field-orientation dependence of the magnetic phase diagram and establish an effective spin Hamiltonian for SrFeO${}_3$ that incorporates a cubic single-ion anisotropy together with bilinear and biquadratic interactions in momentum space, which originate from the spin–charge coupling. In addition, we observe magnetoelastic signatures of a redistribution of the ligand-hole density upon entering the forced ferromagnetic phase. These findings are consistent with an important role of electronic itinerancy associated with the formation of a ligand-hole band in stabilizing multiple-$\yen mathbfQ$ phases.