Because of its strong physical meaning, the decomposition of a symmetric second-order tensor into a deviatoric and a spheric part is heavily used in continuum mechanics. When considering higher-order continua, third-order tensors naturally appear in the formulation of the problem. Therefore researchers had proposed numerous extensions of the decomposition to third-order tensors. But, considering the actual literature, the situation seems to be a bit messy: definitions vary according to authors, improper uses of denomination flourish, and, at the end, the understanding of the physics contained in third-order tensors remains fuzzy. The aim of this paper is to clarify the situation. Using few tools from group representation theory, we will provide an unambiguous and explicit answer to that problem.