The "inverted spectrum argument" was developed as a critique of functionalism. Imagine someone whose color spectrum was inverted: where normal people saw red, this one saw blue, where blue, red. Such a person, raised among normal, English-speaking persons, would be functionally indistinguishable from normal persons: asked to go out to the car and get the blue bag, say, they would perform this task exactly as anyone else would. Neither they nor anyone else would have any way of knowing that their experience of seeing the blue surface of the bag was the same experience the rest of us have when we see a red surface, since they, like everyone else, would refer to such a surface as "blue." Since such a person would be functionally identical to a normal person, a functionalist is committed to the position that there is nothing different about their mental state. But, the argument goes, of course there is something different about their mental state: the quale, or phenomenal quality of the experience, is different. Thus functionalism is false.
This is, I think, the very same argument as the "zombie argument" made famous by David Chalmers: imagine a person who behaved exactly as a normal person does, but who has no conscious experience whatsoever. Again, there would be no way of knowing that one was interacting with a non-conscious "zombie." The arguments can be run using two imagined persons, or one person and a machine. Imagine that I have a wine-identifying device. I put a drop of wine in the device and it spins out the molecules in a centrifuge, and then identifies them using an on-board data base, and has a readout telling me that it is a merlot from such-and-such a vineyard, of such-and-such vintage etc. If I encountered a true wine afficionado I could match him identification for identification, but he would be using his familiarity with respective gustatory qualia whereas I would be using my device. Or imagine an android who was functionally identical to a person but non-conscious. "Inverted spectrum" and "zombie" are two variations of one argument, we can call this the "absent qualia argument." Typically this argument is presented as showing that functionalism (and behaviorism, and operationalist theories of mind in general) founders on the problem of phenomenal properties.
Wittgenstein, for one, noticed that in fact the absent qualia argument demonstrates just the opposite: since it is not even in principle possible for public language (the only kind of language there is, according to Wittgenstein) to pick out private sensations, phenomenal properties are not a problem for operationalist approaches. No theory of mind (or science of mind, or description of mind) will ever include any discussion of the quality of private sensations. These are beyond the range of language.
Wittgenstein deploys the "private language argument" in two different ways. Regarding intentional mental states, he denies the possibility of mental content altogether: there can be no representation, symbolic, isomorphic, or otherwise, in the head. Regarding phenomenal mental states, he does not deny that experience has quality, only that these qualities can be picked out using language. While the problem of intentionality and the problem of phenomenology are, on my view, two distinct metaphysical problems, Wittgenstein can address them both because his thesis is in fact about language, and this thesis can be applied in any number of ways.