We report rearrangement mechanisms and new stationary points for the water tetramer and deduce the associated tunneling splitting patterns and nuclear spin, weights when different processes are assumed to be feasible. The basis sets employed for the ab initio calculations are double-zeta plus polarization (DZP) and DZP with additional diffuse functions (DZP+diff), and results have been obtained within both the Hartree-Fock (HF) and density functional theory frameworks employing the Becke exchange and the Lee-Yang-Parr correlation functionals (BLYP). The results are compared with those found for a relatively sophisticated empirical, rigid-body intermolecular potential. One direct degenerate rearrangement of the cyclic global minimum was characterized in the HF calculations, but disappears when density functional theory is applied. The latter mechanism involves a larger barrier than pathways mediated by higher index saddle points belonging to the torsional space. In principle, doublet splittings could result from tunneling via a number of possible routes, and further calculations will be needed to elucidate the dynamics for this system.