Molecular beam experiments are reported for collisions between oxygen molecules. Total integral cross sections have been measured as a function of the collision energy with the control of molecular alignment. The low collision energy (in the thermal and subthermal range) and the high angular resolution permit observation of the "glory" effect, manifestation of quantum-mechanical interference, which allows an accurate probe of intermolecular interactions. This first complete experimental characterization of the interaction yields a ground (singlet) state bond energy of 17.0 +/- 0.8 meV for the most stable dimer geometry (the two oxygen molecules lying parallel at a distance of 3.56 +/- 0.07 Angstrom). Also the splittings among the singlet, the triplet, and the quintet surfaces are obtained, and a full representation of their angular dependence is reported via a novel harmonic expansion functional form for diatom-diatom interactions. These results indicate that most of the bonding in the dimer comes from van der Waals forces, but chemical (spin--spin) contributions in this open-shell/open-shell system are not negligible (similar to 15% of the van der Waals component of the interaction).