We report X-ray resonant magnetic scattering studies of a Permalloy artificial square ice nanomagnet array, focussing on the field-driven evolution of the sum Σ and difference Δ signals of left and right handed circularly polarized synchrotron X-rays at different lateral positions in reciprocal space Qx. We used X-rays tuned to the Fe L3 resonance energy, with the scattering plane aligned along a principal symmetry axis of the array. Details of the specular Δ hysteresis curve are discussed, following the system magnetization from an initial demagnetized state. The periodic structure gives rise to distinct peaks at in-plane reciprocal Bragg positions, as shown by fitting Σ(Qx) to a model based on a simple unit cell structure. Diffraction order-dependent hysteresis in Δ is observed, indicative of the reordering of magnetization on the system's two interpenetrating sublattices, which markedly deviates from an ideal Ising picture under strong applied fields