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Many molecular "quantum" theories, like "quantum chemistry", conceal that they are actually quantum-classical approaches---they treat one set of molecular degrees of freedom classically while the remaining degrees of freedom follow the laws of quantum mechanics. We show that the prominent "frozen-nuclei approximation", which is often used in molecular control communities, is a further example for such theory reduction: It treats the nuclei of the molecule as classical particles. Here, we demonstrate that the ignorance about the quantum nature of nuclei has far-reaching consequences for the theoretical description of molecules. We analyse the symmetry of oriented and aligned rigid molecules with feasible permutations of identical nuclei and show: The presumption of fixed nuclei corresponds to a localized state that is impossible to create if the existence of stable nuclear spin isomers is a justifiable assumption for the controlled molecule. The results of studies on molecules containing identical nuclei have to be re-evaluated and properly anti-symmetrised, because for such molecules the premise of frozen nuclei is inherently wrong: Molecular wave functions have to obey the spin-statistics theorem twice.