Self-assembled quantum dots are often modeled by continuum models ͑effective mass or k · p͒ that assume the symmetry of the dot to be that of its overall geometric shape. Lens-shaped or conical dots are thus assumed to have continuous cylindrical symmetry C ϱv , whereas pyramidal dots are assumed to have C 4v symmetry. However, considering that the III-V dots are made of atoms arranged on the ͑relaxed͒ positions of a zincblende lattice, one would expect the highest possible symmetry in these structures to be C 2v . In this symmetry group all states are singly degenerate and there are no a priori reason to expect, e.g., the electron P states ͑usually the second and third electron levels of dominant orbital P character͒ to be degenerate. Continuum models, however, predict these states to be energetically degenerate unless an irregular shape is postulated. We show that, in fact, the true ͑atomistic͒ symmetry of the dots is revealed when the effects of ͑i͒ interfacial symmetry, ͑ii͒ atomistic strain, and ͑iii͒ piezoelectricity are taken into account. We quantify the contributions of each of these effects separately by calculating the splitting of electron P levels for different dot shapes at different levels of theory. We find that for an ideal square-based pyramidal InAs/GaAs dot the interfacial symmetry of the unrelaxed dot splits the P level by 3.9 meV, atomistic relaxation adds a splitting of 18.3 meV ͑zero if continuum elasticity is used to calculate strain͒ and piezoelectricity reduces the splitting by Ϫ8.4 meV, for a total splitting of 13.8 meV. We further show that the atomistic effects ͑i͒ and ͑ii͒ favor an orientation of the electron wave functions along the ͓110͔ direction while effect ͑iii͒ favors the ͓110͔ direction. Whereas effects ͑i͒ ϩ ͑ii͒ prevail for a pyramidal dot, for a lens shaped dot, effect ͑iii͒ is dominant. We show that the 8-band k · p method, applied to pyramidal InAs/GaAs dots describes incorrectly the splitting and order of P levels ͑-9 meV instead of 14 meV splitting͒ and yields the orientation ͓110͔ instead of ͓110͔.