We report the determination of two dimer water potential energy surfaces via direct inversion of spectroscopic data. The first surface, rigid, employs the MCY functional form originally fitted by Clementi and co-workers from ab initio calculations, modified by adjunction of a fifth, uncharged, site to improve the dispersion component. The vibration-rotation-tunneling energy levels were computed by means of the pseudospectral split Hamiltonian method that we developed previously. The fitted surface shows considerable improvement as compared to the original one: transitions among the ground-state manifold are in error by at most 0.2 cm(-1), and excited state band origins (up to 150 cm(-1)) are reproduced to within 0.5 to 3 cm(-1). For the second surface, flexible, we used the same modified MCY functional form, considered now to depend on the 12 internal degrees of freedom, and augmented by the monomer potential energy terms. The water dimer is described in its full dimensionality by collision-type coordinates in order to access the whole configuration sampled by this floppy system. Internal motions of the monomers (stretches and bends) are explicitly considered by invoking an adiabatic separation between the slow (intermonomeric) and fast (intramonomeric) modes. This (6+6)d adiabatic formulation allows us to recast the calculations into an equivalent six-dimensional dynamics problem (similar topseudorigid monomers) on an effective potential energy surface. The resulting, fitted, fully flexible dimer potential leads to a much better agreement with experiment than does the rigid version, as examplified by the standard deviation on all observed frequencies being reduced by a factor of 3. It is shown that monomer flexibility is essential in order to reproduce the experimental transitions.