The adsorption of phthalocyanines (Pc) to various surfaces has recently been reported to lead to a lowering of symmetry from C4 to C2 in scanning tunneling microscope (STM) images. Possible origins of the reduced symmetry involve the electronic structure or geometric deformation of the molecules. Here, the origin of the reduction is clarified from a comprehensive theoretical study of CoPc adsorbed on the Cu(111) surface along with the experimental STM data. Total energy calculations using different schemes for the exchange-correlation energy and STM simulations are compared against experimental data. We find that the symmetry reduction is only reproduced when van der Waals corrections are included into the formalism. It is caused by a deformation along the two perpendicular molecular axes, one of them coming closer to the surface by around 0.2 Å. An electronic structure analysis reveals (i) the relevance of the CoPc interaction with the Cu(111) surface state and (ii) that intramolecular features in dI/dV maps clearly discriminate a Co-derived state from the rest of the Pc states.