A combined spectroscopic and theoretical investigation is devoted to the vibrational fine structure in the broad band spectrum at 4.2 K of the T-1(1g) and T-1(2g) states of Rh(III) d(6) doped in NaCl. The vibrational structure is resolved with a noise reduction technique using quadrature mirror filters. A weak signal at 16805 cm-(1) is identified as the zero phonon line of the T-1(1g) <-- (1)A(1g) transition, 194 cm-(1) below a false origin of a progression in a single 264 cm-(1) mode. The observed spectrum can be reproduced quantitatively by ab initio calculations involving identification of the Jahn-Teller active distortions, derivation of the shapes of excited state potential energy surfaces, and calculation of the Franck-Condon factors. The dominant mode of the progression is identified as a combination of the a(1g) and e(g)(a) modes with a calculated frequency of 279 cm-(1). The calculations further reveal the existence of a conical intersection between the T-1(1g) and T-1(2g) states, situated in the optical window between the two absorption bands and expected to play an important role in the deactivation of the T-1(2g) state.