The global and local minima, stationary points, and torsional rearrangement processes of cyclic homodromic (H2O)(4) were studied on its four-dimensional torsional intermolecular potential energy surface. Eight different energetically low-lying torsional stationary point structures were found by ab initio theory, and fully structure-optimized at the second-order Moller-Plesset level, using large basis sets. Second-order energies close to the one-particle basis set limit were obtained at these geometries using the explicitly correlated Moller-Plesset method. The effects of higher-order correlation energy terms were investigated by coupled cluster theory, and terms beyond second order were found to cancel in good approximation. The S-4 symmetric global minimum has a square and almost planar O...O...O...O arrangement with free O-H bonds alternating "up" and "down" relative to this plane, with two isometric versions of this structure. Another torsional conformer with two neighboring up O-H bonds followed by two neighboring down O-H bonds is a local minimum, 0.93 kcal/mol above the global minimum. The four versions of this structure are connected to the global minima via two distinct but almost degenerate first-order torsional saddle points, which occur as two sets of eight isometric versions each, both about 1.24 kcal/mol above the global minimum. Yet another set of eight second-order saddle points lies at 1.38 kcal/mol. The structure with three O-H bonds up and one down is not a stationary point, while the structure with all four O-H bonds on the same side of the plane is a first-order saddle point. The fully planar C-4h symmetric structure is a fourth-order stationary point 2.8 kcal/mol above the minimum. The torsional interconversion paths between this multitude of points are complex, and are discussed in three-dimensional spaces of symmetry-adapted torsional coordinates, and also in a network representation. The torsional normal-mode eigenvectors point fairly directly along the torsional interconversion pathways, but the harmonic frequencies are well below the corresponding barriers. Tunneling interconversion between torsional conformers is, hence, less important than for the water trimer.