The fundamental Debye-Stokes-Einstein (DSE) relation between rotational relaxation times and shear viscosity attracts longstanding research interest as one of the most important characteristics of many glass-forming liquids. Here, we provide strong evidence, missing so far, for the relevance of anisotropy for DSE-related behavior. Dielectric spectroscopy and shear viscosity measurements were employed to get insight into the decoupling between reorientation relaxation times and viscosity for anisotropic glass-formers with dipole moments oriented parallel or perpendicular to the long molecular axis. We found that in the case of large and anisotropic molecules, the breakdown of DSE relation depends on the component of anisotropic rotation contributing to the dielectric response. Specifically, for glass-formers with dipole moment perpendicular to the long molecular axis, the DSE relation was found to be valid throughout the supercooled regime. Contrary, a departure from the DSE predictions in the intermediate supercooled regime, was observed for glass-formers where only short-axes contributions were relevant in dielectric response. MD simulations revealed differences in the mechanism of short and long axes reorientations suggesting that for anisotropic objects, not the reorientation mechanism itself, but the aspect of anisotropic motion, is the key to understanding the behavior of these systems in the context of DSE relation.