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We investigate the electronic and structural properties of CuO, which shows significant deviations from the trends obeyed by other transition-metal monoxides. Using an extended Hubbard-based corrective functional, we uncover an orbitally ordered insulating ground state for the cubic phase of this material, which was expected but, to the best of our knowledge, was not found in the literature. This insulating state results from a fine balance between the tendency of Cu to complete its -shell and Hund's rule magnetism. Starting from the ground state for the cubic phase, we also study tetragonal distortions of the unit cell (recently reported in experiments) and identify the equilibrium structure. Our calculations reveal an unexpected richness of possible magnetic and orbital orders, relatively close in energy to the ground state, whose stability depends on the sign and nature of distortion.