The low-energy bifurcation tunnelling mechanism for the water trimer is characterized at various levels of theory, ranging from empirical rigid-body effective pair potentials to ab initio calculations including corrections for the correlation energy. The resulting tunnelling splitting patterns and effective molecular symmetry groups are deduced and compared with experimental results. We find that the number of intrinsic flips associated with this mechanism is very sensitive to the level of theory employed to calculate the pathway. However, the six distinct bifurcation paths only yield two possible splitting patterns, both of which include quartets. When non-degenerate tunnelling via the crown minimum is admitted into the picture the effective molecular symmetry group and the splitting patterns of the original levels are unchanged. We conclude that delocalisation of the wavefunctions over the crown minima is unlikely to have a significant effect, and that the quartet splittings found experimentally are probably due to the bifurcation mechanism.