For decades, protonated methane, CH(5)(+), has provided new surprises and challenges for both experimentalists and theoreticians. This is because of the correlated large-amplitude motion of its five protons around the carbon nucleus, which leads to so-called hydrogen scrambling and causes a fluxional molecular structure. Here, the infrared spectra of all its H/D isotopologues have been measured using the 'Laser Induced Reactions' technique. Their shapes are found to be extremely dissimilar and depend strongly on the level of deuteration (only CD(5)(+) is similar to CH(5)(+)). All the spectra can be reproduced and assigned based on ab initio quantum simulations. The occupation of the topologically different sites by protons and deuterons is found to be strongly non-combinatorial and thus non-classical. This purely quantum-statistical effect implies a breaking of the classical symmetry of the site occupations induced by zero-point fluctuations, and this phenomenon is key to understanding the spectral changes studied here.