What [the identity theorist] says is that there is only ONE thing there, the brain and its various states, and you cannot reduce something to itself!
He goes on to contrast this with linguistic, theoretical reductions, and explains how those are irrelevant to the debate between physicalists and dualists. I agree with that part. But I think he's wrong to think that linguistic reductions are the only coherent form of reduction.
We can see this because, as I've been saying all along, the question is whether qualia are reducible in the same sense that tables and chairs are, whatever that may be. Now, it's an open question whether our talk of tables and chairs could be replaced by (perhaps complicated and long-winded) talk purely in the language of microphysical theory. But we don't care about talk. What matters is that the facts about tables are obviously settled by the microphysical facts. If you have a coarse-grained conception of 'facts', maybe they are even one and the same fact. Even so, we can get to a metaphysical notion of reduction by appeal to the truthmakers for our sentences. Regardless of whether table talk is linguistically replaceable by particle talk, there's no question that the microphysical facts are what make our table statements true (if they are true).
Once you've included the microphysical facts in your base facts, you do not need to add any further 'table facts' in addition. Those are already covered. It is in this sense that table facts are reducible to physical facts. And it is in this sense that the question of physicalism comes down to the question whether qualia are reducible. It is simply the question whether we need to add phenomenal facts to our fundamental base facts, or whether they "come along for free" (like tables do) given the physical facts P.
(I find it convenient to use the term 'reducible' to invoke this idea, but you're of course free to pick another word if you prefer. What's not helpful is to simply insist, "the debate between the dualist and the materialist is in no way a debate about reduction", and so ignore my underlying idea concerning what the debate is about. RB wanted to focus on what counts as 'physical' or 'non-physical', but that soon degrades into semantics. The substantive issue, as I see things, is whether we must include qualia as an additional primitive among the base facts. This understanding makes it clear why RB's "non-physical zombie" parody argument falls flat (to put it mildly). See my 'Zombie Review' for more detail.)