Most papers on methyl group tunnelling use a model in which the motion of a rigid rotator is hindered by a potential of threefold symmetry. There are a number of difficulties with such a description and, in this paper, the model is replaced by one which describes the motion of three protons, which have mutual interactions and which are hindered by an external potential. The analysis makes considerable use of symmetry ideas and arrives at conclusions which are substantially different from those of the previous model. A comparison of the two models shows that, in the absence of dipolar interactions, both will have an orbital singlet level with spin of 3/2, but that the level described as 2E in the rigid-rotator model will not occur in the present model and that there will be no corresponding level with fourfold degeneracy. Instead there will be two separated orbital singlets, each with spin 1/2. The inclusion of the dipolar interaction splits the quartet into two doublets, so that finally there are four separated low-lying energy levels, each being doubly degenerate.