The lack of structural symmetry which usually characterizes semiconductor quantum dots lifts the energetic degeneracy of the bright excitonic states and hampers severely their use as high-fidelity sources of entangled photons. We demonstrate experimentally and theoretically that it is always possible to restore the excitonic degeneracy by the simultaneous application of large strain and electric fields. This is achieved by using one external perturbation to align the polarization of the exciton emission along the axis of the second perturbation, which then erases completely the energy splitting of the states. This result, which holds for any quantum dot structure, highlights the potential of combining complementary external fields to create artificial atoms meeting the stringent requirements posed by scalable semiconductor-based quantum technology.