A new process associated with the nonlinear optical properties of the electromagnetic quantum vacuum is described. It corresponds to the superradiant emission of photons, resulting from the interaction of an intense laser pulse with frequency ω0 with a counter-propagating high-harmonic signal with a spectrum of frequencies nω1, for n integer, in the absence of matter. Under certain conditions, photon emission from vacuum will be enhanced by the square of the number of intense spikes associated with the high-harmonic pulse. This occurs when the field created by the successive spikes is coherently emitted, as in typical superradiant processes involving atoms. Subradiant conditions, where the nonlinearity of quantum vacuum is entirely suppressed, can equally be defined.