Third-harmonic scattering is a nonlinear optical process that involves the molecular second-hyperpolarizability, gamma. This work presents a rigorous quantum electrodynamical analysis of the scattering process, involving a partially index-symmetric construction of the fourth-rank gamma tensor dispensing with the Kleinman symmetry condition. To account for stochastic molecular rotation in fluids, methods of isotropic averaging must be employed to relate the molecular properties to accessible experimental quantities such as depolarization ratio. A complete eighth-rank tensor rotational average yields results for observable third-harmonic scattering rates, cast as a function of the natural-invariant gamma components, and the polarization geometry of the experiment. Decomposing the tensor gamma into irreducible weights allows specific predictions to be made for each molecular point group, allowing greater discrimination between the results for different molecular symmetries.