Quantum dots (QDs) of high symmetry (e.g., C3v) have degenerate bright exciton states, unlike QDs of C2v symmetry, making them intrinsically suitable for the generation of entangled photon pairs. Deviations from C3v symmetry are detected in real QDs by polarization-resolved photoluminescence spectroscopy in side-view geometry of InGaAs/AlGaAs dots formed in tetrahedral pyramids. The theoretical analysis reveals both an additional symmetry plane and weak symmetry breaking, as well as the interplay with electron-hole and hole-hole exchange interactions manifested by the excitonic fine structure.