Using proton and deuterium NMR spectroscopy, this work provides a description of the molecular orientational order of the mesogenic groups in both the N-tb and the classical nematic phase (N) of the liquid crystal dimer CB-C9-CB. The proton NMR data were collected at high field (7 T) with the nematic domains aligned by the field. Deuterium NMR data obtained from aligned samples at 11.7 T, published by Hoffmann, A. et al. Soft Matter 2015, 11, 850, were also considered in this study. Using the first-order perturbation theory, we carried out detailed simulations of the proton spectra from the terminal mesogenic groups along with the quadrupolar splittings from the carbon-deuterium bonds in the first chain positions, which allow for the determination of the Saupe order tensor for the mesogenic groups. We show that the main mechanism that induces the change of the H-1 NMR spectrum and the quadrupolar splittings at the N-N-tb phase transition is the change of the orientation of the most ordered molecular frame (eigenframe of the Saupe tensor), along with the onset of the molecular biaxiality parameter, D. This orientation change is associated with the achiral symmetry breaking at the N-N-tb phase transition.