X-ray magnetic circular dichroism (XMCD) measurements have been performed at the Cr and Fe K-edges for the two-dimensional oxalate-based magnet {[N(C4H9)(4)][(FeFe0.77Cr0.23III)-Fe-II-Cr-III(ox)(3)]}(n) (ox = C2O42-) in its long-range ordered magnetic state. First, the XMCD spectra recorded under a 2 T applied magnetic field are used to probe the local exchange interaction between the metal ions through the relative orientation of their spins in the magnetic field. The remnant local magnetic moments on the Cr and Fe ions have been clearly evidenced, analyzed, and compared to the total remnant magnetization measured by conventional magnetometry. The respective impact of single ion anisotropy, magneto-structural anisotropy, and exchange interaction on the remnant magnetic state is then discussed.