Atomistic and quantum DFT calculations are carried out for models of Ce0.5Zr0.5O2 mixed oxide having different cation orderings within a fluorite-type structure. The DFT results are more accurate than those given by the atomistic calculations; these latter exaggerate oxygen sublattice distortions and energy differences between different cation orderings. The results imply that using an "averaged cations" approach in atomistic simulations is likely to miss relevant effects of cation distribution details at local levels. The lowest energy is obtained with a double-layer type of cation clustering, in agreement with some EXAFS data; the typical tetragonal crystal cell distortion, however, is obtained optimally with a different, nonclustered scheelite-type ordering, similar to that of other related mixed oxides, which is also among the lowest energy structures obtained and agrees with other published EXAFS data. Implications of the predicted Zr-O coordination configurations for the interpretation of EXAFS data are discussed. The K-phase, formed by controlled reoxidation of the Ce2Zr2O7 pyrochlore so as to keep the same cation arrangement as in the latter, is effectively confirmed to be rather less stable, at least when defect-free. The DFT calculations predict for it a crystal symmetry lower than cubic but with almost exactly cubic unit cell dimensions; its discrepancies with the XRD-derived published crystal structure are discussed in connection with symmetry lowering effects and with the likelihood of the presence of anionic Frenkel defects. The possibility is discussed that this phase could have different surface acidity and XPS features (because of modified Ce ion charges) as well as a smaller band gap in agreement with some experimental observations.