Mean square displacements of hydrogen atoms in glass-forming materials and proteins, as reported by incoherent elastic neutron scattering, show kinks in their temperature dependence. This crossover, known as the dynamical transition, connects two approximately linear regimes. It is often assigned to the dynamical freezing of subsets of molecular modes at the point of equality between their corresponding relaxation times and the instrumental observation window. The origin of the dynamical transition in glass-forming glycerol is studied here by extensive molecular dynamics simulations. We find the dynamical transition to occur for both the center of mass translations and the molecular rotations at the same temperature, insensitive to changes of the observation window. Both the translational and rotational dynamics of glycerol show a dynamic crossover from the structural to a secondary relaxation at the temperature of the dynamical transition. A significant and discontinuous increase in the orientational Kirkwood factor and in the dielectric constant is observed in the same range of temperatures. We, however, do not find any indications of a true thermodynamic transition to an ordered low-temperature phase. We therefore suggest that all observed crossovers are dynamic in character. The increase in the dielectric constant is related to the dynamic freezing of dipolar domains on the time-scale of simulations.