It is generally agreed upon that qualities of conscious experience instantiate structural properties, usually called relations. They furnish a representation of qualities (or qualia, in fact) in terms of a mathematical space Q (rather than a set), which is crucial to both modelling and measuring of conscious experience." "What is usually disregarded is that “only such structural properties generalize across individuals” (Austen Clark), but that qualities themselves as differentiated by stimulus specifications, behavior or reports do not. We show that this implies that only the part of Q which is invariant with respect to the automorphism group has a well-defined referent, while individual elements do not. This poses a prima facie limitation of any theory or experiment that aims to address individual qualities. We show how mathematical theories of consciousness can overcome this limitation via symmetry groups and group actions, making accessible to science what is properly called private language.