Labels are fun, and get attention. They can be a useful shorthand to capture an idea, or related set of ideas. But is there really a New Old Keynesian school of thought? I don’t think so. Here are a couple of bold assertions, which I think I believe, and which I will try to justify. First, in academic research terms there is only one meaningful division, between mainstream and heterodox. (Of course the heterodox divide themselves up into various ‘schools’, but their size is small and their influence is also small.) Second, in macroeconomic policy terms I think there is only one meaningful significant division, between mainstream and anti-Keynesians.
But before trying to justify these statements, I want to defend being a killjoy. As I said, putting people into categories can be fun - why spoil it by taking the exercise seriously? Two reasons. First, I want to make some points which do not get said often enough on economics blogs. Second, labels can lead to confusion or worse. Just think about the label Keynesian. Any sensible definition would involve the words sticky prices and aggregate demand. Yet there are still some economists (generally not academics) who think Keynesian means believing fiscal rather than monetary policy should be used to stabilise demand. Fifty years ago maybe, but no longer. Even worse are non-economists who think being a Keynesian means believing in market imperfections, government intervention in general and a mixed economy. (If you do not believe this happens, look at the definition in Wikipedia.)
So what do I mean by a meaningful division in academic research terms? I mean speaking a different language. Thanks to the microfoundations revolution in macro, mainstream macroeconomists speak the same language. I can go to a seminar that involves an RBC model with flexible prices and no involuntary unemployment and still contribute and possibly learn something. Equally an economist like John Cochrane can and does engage in meaningful discussions of New Keynesian theory (pdf). 
Of course all academic macroeconomists have their own idea of how the world actually works and will probably do research using models that roughly conform to that. Yet I think you would be hard put to draw meaningful boundaries here. Take John Quiggin’s Old/New Keynesian post, for example (which followed this from Tyler Cowen). He characterises New New Keynesians as those still working with DSGE models who are now attempting to add financial frictions. He wants to argue (and labels New Old Keynesian) the idea that following this recession, there “is no unique long-run equilibrium growth path, determined by technology and preferences, to which the economy is bound to return. In particular, the loss of productive capacity, skills and so on in the current depression is, for all practical purposes, permanent.” Now listening with my mainstream ears, this sounds like a combination of hysteresis effects and endogenous growth, which sounds interesting. Yet I also think we can learn a lot from adding financial frictions to DSGE models. Does this make me a middle aged Keynesian?
What I suspect Quiggin is getting at here is that New New Keynesians are still following a microfoundations research programme (using DSGE), whereas he would not. Now many mainstream macroeconomists, myself included, can be pretty critical of the limitations that this programme can place on economic thinking, particularly if it is taken too literally by microfoundations purists. But like it or not, that is how most macro research is done nowadays in the mainstream, and I see no sign of this changing anytime soon. (Paul Krugman discusses some reasons why here.) My own view is that I would like to see more tolerance and a greater variety of modelling approaches, but a pragmatic microfoundations macro will and should remain the major academic research paradigm.
When it comes to macroeconomic policy, and keeping to the different language idea, the only significant division I see is between the mainstream macro practiced by most economists, including those in most central banks, and anti-Keynesians. By anti-Keynesian I mean those who deny the potential for aggregate demand to influence output and unemployment in the short term.  Why do I use the term anti-Keynesian rather than, say, New Classical? Partly because New Keynesian economics essentially just augments New Classical macroeconomics with sticky prices. But also because as far as I can see what holds anti-Keynesians together isn’t some coherent and realistic view of the world, but instead a dislike of what taking aggregate demand seriously implies.
What is incoherent about believing in pretty flexible prices, you might ask? Two things. First, as I have argued before, with the demise of the Pigou effect flexible prices do not get you out of a deficient demand problem at the zero lower bound when there are inflation targets. Second, the evidence that prices are not flexible is so overwhelming that you need something else to drive you to ignore this evidence. Or to put it another way, you need something pretty strong for politicians or economists to make the ‘schoolboy error’ that is Says Law, which is why I think the basis of the anti-Keynesian view is essentially ideological.
Of course there are a huge number of policy debates in macroeconomics, and you can attach labels to those if you like. Should we use fiscal stimulus at the zero lower bound, for example. Was austerity a good idea? However, anti-Keynesians aside, I don’t think these debates reveal large fault lines in economic thinking. Economists do not rigidly line up on one side or another, and some even change their mind over time as the facts change. It is possible to have serious discussions about the effectiveness of monetary policy, the dangers of high debt etc. The only group where a discussion can fail to get off the ground is with those who contend that aggregate demand is always irrelevant.
 Heterodox economists might argue that they have to be bilingual - they are able to speak mainstream, even if they prefer not to among friends. Those more critical might detect a reluctance to get past certain words.
 An alternative, and more positive, way to define the anti-Keynesian group is that they believe macroeconomic outcomes are essentially efficient, and so intervention by a government (or central bank, beyond providing a nominal anchor) is not required. This difference might be important in placing someone like Roger Farmer (who I’m glad to see now has a blog), who is not an anti-Keynesian under this positive definition, but might be using my more negative criteria.