Temperature-dependent Raman scattering and differential scanning calorimetry were applied to the study of the hybrid organic-inorganic azide-perovskite [(CH3)4N][Cd(N3)3], a compound with multiple structural phase transitions as a function of temperature. A significant entropy variation was observed associated to such phase transitions, |∆S| ~ 62.09 J·kg−1 K−1, together with both a positive high barocaloric (BC) coefficient |δTt/δP| ~ 12.39 K kbar−1 and an inverse barocaloric (BC) coefficient |δTt/δP| ~ −6.52 kbar−1, features that render this compound interesting for barocaloric applications. As for the obtained Raman spectra, they revealed that molecular vibrations associated to the NC4, N3– and CH3 molecular groups exhibit clear anomalies during the phase transitions, which include splits and discontinuity in the phonon wavenumber and lifetime. Furthermore, variation of the TMA+ and N3– modes with temperature revealed that while some modes follow the conventional red shift upon heating, others exhibit an unconventional blue shift, a result which was related to the weakening of the intermolecular interactions between the TMA (tetramethylammonium) cations and the azide ligands and the concomitant strengthening of the intramolecular bondings. Therefore, these studies show that Raman spectroscopy is a powerful tool to gain information about phase transitions, structures and intermolecular interactions between the A-cation and the framework, even in complex hybrid organic-inorganic perovskites with highly disordered phases.