The theory of vibrational hyper‐Raman scattering (inelastic second harmonic light scattering) is developed within the framework of quantum electrodynamics. The dynamical system comprises the molecule and the radiation field, coupled by multipolar interactions. In the present work, the electric dipole approximation is employed and the results involve the hyperpolarizability tensor. In contrast to previous work, full index symmetry of this tensor is not assumed. The tensor is decomposed into irreducible weights forming the basis for a new vibrational mode classification scheme. A set of five experiments is specified which allows unambiguous assignment of an active mode to one of six classes. The scheme has been applied to several molecules and compared with some experimental results. Explicit expressions for scattering intensities for a wide variety of experimental configurations are given; results for depolarization ratios are also presented.