The local structure of the nonplanar phthalocyanine, vanadyl phthalocyanine (VOPc), adsorbed on Cu(111) at a coverage of approximately one-half of a saturated molecular layer, has been investigated by a combination of normal-incidence X-ray standing waves (NIXSW), scanned-energy mode photoelectron diffraction (PhD), and density-functional theory (DFT), complemented by scanning tunnelling microscopy (STM). Qualitative assessment of the NIXSW data clearly shows that both "up" and "down" orientations of the molecule (with V=O pointing out of, and into, the surface) must coexist on the surface. O 1s PhD proves to be inconclusive regarding the molecular orientation. DFT calculations, using two different dispersion correction schemes, show good quantitative agreement with the NIXSW structural results for equal co-occupation of the two different molecular orientations and clearly favor the many body dispersion (MBD) method to deal with long-range dispersion forces. The calculated relative adsorption energies of the differently oriented molecules at the lowest coverage show a strong preference for the "up" orientation, but at higher local coverages, this energetic difference decreases, and mixed orientation phases are almost energetically equivalent to pure "up"-oriented phases. DFT-based Tersoff-Hamann simulations of STM topographs for the two orientations cast some light on the extent to which such images provide a reliable guide to molecular orientation.