A nonlinear optical effect in which a linearly polarized wave propagating in a single quadratic medium is converted into a wave that is cross polarized to the input wave is investigated theoretically and observed experimentally in β-barium borate crystal. It is proved that this effect is a result of cascading of two different second-order processes. It starts with the generation of an extraordinary second-harmonic wave by type I interaction and is followed by type II difference-frequency mixing between the second-harmonic wave and the ordinary fundamental wave. The experiment was performed (a) for phase-matched type I interaction and non-phase-matched type II interaction and (b) for non-phase-matched type I interaction and phase-matched type II interaction. The observed generation of a cross-polarized wave is to our knowledge the only cubic effect whose first manifestation has been observed in quadratic crystal.