  Up to now, experiments involving M\"ossbauer nuclei driven by x-rays have
been restricted to the low-excitation regime. Here, a setup is proposed which
promises significant excitation, ideally exceeding full inversion of the
nuclear ensemble, at x-ray light sources under construction. We further
introduce a method to experimentally verify such inversions, in which
population inversions manifest themselves in symmetry flips of suitably
recorded spectra. It neither requires per-shot spectra of the incoming x-ray
pulses, nor absolute measurements of the scattered light intensity.
