The family of compounds CBrnCl4-n has been proven helpful in unraveling microscopic mechanisms responsible for glassy behavior. Some of the family members show translational ordered phases with minimal disorder which appears to reveal glassy features, thus deserving special attention in the search for universal glass anomalies. In this work, we studied CBrCl3 dynamics by performing extensive molecular dynamics simulations. Molecules of this compound perform reorientational discrete jumps, where the atoms exchange equivalent positions among each other revealing a cage-orientational jump motion fully comparable to the cage-rototranslational jump motion in supercooled liquids. Correlation times were calculated from rotational autocorrelation functions showing good agreement with previous reported dielectric results. From mean waiting and persistence times calculated directly from trajectory results, we are able to explain which microscopic mechanisms lead to characteristic times associated with α- and β-relaxation times measured experimentally. We found that two nonequivalent groups of molecules have a longer characteristic time than the other two nonequivalent groups, both of them belonging to the asymmetric unit of the monoclinic (C2/c) lattice.