Jonathan Haidt mostly gets it about liberals and conservatives. He comments on libertarians as well, and what he reports seems about right, but I don't feel positioned to be quite as definite about that.
Brief Review. You can read in detail for yourself at his Moral Foundations site. I do not recommend his TED talk from 2008, as it predates his important revisions and will muddy the waters. You can also skim what I have written over the years.
Liberals operate from three unequal but solid strands in their morality: Care vs. Harm is strongest; Liberty vs Oppression (an axis that was not in his original matrix); Fairness vs. Cheating is third.
Conservatives add three others into their determination of whether something is moral or not, and Haidt claims all six are about equal in strength: Loyalty vs. Betrayal; Authority vs. Subversion; Sanctity versus Degradation.
Libertarians have one massive axis and one minor one. Liberty vs Oppression dominates the field, with Fairness vs Cheating chiming in.
I claimed right from the start when I first mentioned him in 2008 that liberals do, in fact, use the remaining three categories, it just wasn't picked up in his questioning. The authorities mentioned were conservative ones , the "disgusts/sacrals" left out GMO foods, hunting (and here), environmental problems of low practical but high aesthetic concern, a good percentage of vegetarianism, iconic liberals . Plus my favorite, about the healing power of John Lennon's piano. Because clearly, those must be conservatives driving that one.
Black conservatives can tell you about the loyalty/betrayal axis.
Haidt does acknowledge in passing that liberals do actually respond in these ways, and even uses a couple of the examples (Gandhi, MLK, and as above) I have been hammering all these years. (I'm taking full credit for all of that.) Yet in the end he discounts that they are much of a factor. I continue to say he's just wrong in this. However, I think it is a fair point that Conservatives do respond to these more as well as differently, so his central point remains important.
He remains a partisan Democrat/Liberal and is quite up front about it. He devotes occasional sections to advising Democrats what they should do to win elections. (Mostly, they seem not to do this. A great many remain so incensed at what they view as a betrayal that they cannot hear him. Not only the comments sections, but discussions from other academics excoriate him - a social psychologist -for not recognising that conservatives are simply morally inadequate, stupid, and evil. Which I think is evidence for my point that Loyalty/Betrayal is active for liberals.) Recently an academic paper and an essay of his have been circulating about ideological intolerance by liberals starting well before college, citing talks he has been giving at private high schools. He doesn't think academia leaning left is a problem. He does think that academia being entirely leftist is a huge problem. He thinks intellectual diversity promotes better thinking and research.
Yet when he summarises the liberal and conservative narratives late in The Righteous Mind he is nearly fair. There is a whiff of dismissal of conservative thought (the defense of what is good) which he is unaware of, but not too terrible. I am more concerned about his summary of liberal narrative (the progression of rights). It seemed spot on at first, but he seems to forget his own main point about the rider and the elephant: he attributes their moral view to the rider. Throughout the book he teaches, and gives good illustrations, that everyone is 90% elephant and 10% rider. At heart, I think he believes the rider percentage is higher for liberals.
I mostly breezed by that "religion is important for internal bonding and enhances group selection" part of the book in an earlier post. Haidt spends a fair bit of time on it in the book. He is an atheist that now hangs out with UU's I recall, for exactly that reason. Do we need to discuss that further? I don't have much to add, but it might be a fun discussion. Maybe an open thread in a day or two. Though I don't know if this audience likes to do that.