Hamiltonian is derived for the rotating water trimer with three internal motions-the rotations of the monomers about their hydrogen bonds. We obtain an expression of the kinetic energy operator, which is a non-trivial extension of earlier heuristic forms used for the non-rotating trimer. The Coriolis coupling operator between the single-axis monomer angular momenta and the overall trimer rotation is given for the first time. To analyze the effects of the tunneling and Coriolis splittings on the energy levels of the trimer, we introduced a qualitative model for the pseudo-rotation and donor tunneling. By perturbation theory and application of the permutation-inversion groups G(6) and G(48) we obtain algebraic expressions for the splittings due to pseudo-rotation and donor tunneling, respectively. The pseudo-rotation does not produce any internal angular momentum and does not yield first order Coriolis splitting, but in second order the Coriolis coupling lifts various degeneracies and gives rise to observable J-dependent splittings. Donor tunneling splits every pseudo-rotation level into a quartet and those levels in this quartet that belong to the three-dimensional irreps of G(48) into doublets. For J>0 a rather complex pattern of larger (for the internal states with G(6) labels k=+/-1 and +/-2) and smaller (for the levels with k=0 and k=3) splittings is obtained, especially for the substates with K=1 which are Coriolis coupled to the K=0 substates. The results of calculations in the companion paper, together with the model introduced in the present paper, will be used to interpret all the tunneling splittings observed in high-resolution spectra of (H2O)(3) and (D2O)(3).