A new potential energy surface (PES) for the quintet state of rigid O(2)((3)Sigma(g)(-)) + O(2)((3)Sigma(g)(-)) has been obtained using restricted coupled-cluster theory with singles, doubles, and perturbative triple excitations [RCCSD(T)]. A large number of relative orientations of the monomers (65) and intermolecular distances (17) have been considered. A spherical harmonic expansion of the interaction potential has been built from the ab initio data. It involves 29 terms, as a consequence of the large anisotropy of the interaction. The spherically averaged term agrees quite well with the one obtained from analysis of total integral cross sections. The absolute minimum of the PES corresponds to the crossed (D(2d)) structure (X shape) with an intermolecular distance of 6.224 bohrs and a well depth of 16.27 meV. Interestingly, the PES presents another (local) minimum close in energy (15.66 meV) at 6.50 bohrs and within a planar skewed geometry (S shape). We find that the origin of this second structure is due to the orientational dependence of the spin-exchange interactions which break the spin degeneracy and leads to three distinct intermolecular PESs with singlet, triplet, and quintet multiplicities. The lowest vibrational bound states of the O(2)-O(2) dimer have been obtained and it is found that they reflect the above mentioned topological features of the PES: The first allowed bound state for the (16)O isotope has an X structure but the next state is just 0.12 meV higher in energy and exhibits an S shape.