Early measurements of the binding energy of CO/MgO(001) gave low values, 0.15-0.17 eV, for the regular site, while later experiments using two independent experimental techniques (temperature programmed desorption and Clausius-Clapeyron plots) have given a much higher value, 0.43-0.45 eV. Theory has shown the opposite trend : early calculations gave a value of 0.38 eV, while the latest results are 0.07-0.11 eV. We have performed a careful theoretical analysis of the Clausius-Clapeyron experiment which is found to be in the assumed equilibrium region. Quantum chemical cluster modelling of regular, step, and corner sites show that only a low-coordinated (corner) site can fit both the measured binding energy and vibrational frequency shift. Accurate embedding techniques with full account of the crystal potential and a high-level treatment of dynamical correlation using large basis sets have been used. Effects of relaxation of the step have been investigated using atomistic simulation techniques. The binding energies (corrected for basis set superposition errors) are computed as 0.08, 0.18, and 0.48 eV for regular, step and corner sites, respectively. Theoretical temperature programmed desorption spectra have been generated and show that desorption from regular and step sites should be expected at 40 and 80 K, respectively. At a corner site the molecule is tilted at 45 degrees to the normal. From angle-resolved near-edge extended x-ray absorption fine structure spectroscopy of CO/NiO(100) the molecule has been found to be vertical. Theoretical spectra have been generated including averaging over the frustrated rotations; agreement with the experimental intensity distribution is obtained also for the tilted (at 0 K) geometry. It is suggested that the oxide films contain a high density of defects.