Detailed analysis of time-resolved electron paramagnetic resonance (EPR) line shapes of photoexcited triplet states of C60 in isotropic and anisotropic matrices has been carried out. Two main motional models have been tested over a very wide temperature range. For most temperatures, the best fit to the experimental line shapes was obtained with a model of discrete jumps between two configurations, which can be formally defined as an exchange between the longitudinal (Z) dipolar axis and one of the two transverse axes (X,Y). This exchange process is probably related to two distorted Jahn-Teller (JT) configurations of the molecule. For very high temperatures, the best fit was obtained with a recently developed model of Brownian rotational diffusion. In the latter cases, relating to the liquid phase of the solvent, the triplets appear to be in thermal spin equilibrium, and their kinetics can be described by a special form of transient nutations. In the former cases, related to the glass and amorphous phases (toluene) or nematic phase (E-7), the triplets are clearly spin polarized. For very low temperatures the motional (jump) rate is practically temperature independent. This seems to be due to the nature of the motion, i.e., exchange between JT configurations, rather than actual molecular motion.