The main difference between left and right with regard to property rights is simply that the right is invested in a lot of rhetoric about markets and property rights and the left is invested in different historical and rhetorical tropes.The exact same thing is true of issues surrounding federalism, right? In reality, both Democrats and Republicans tend to support states when it is otherwise convenient to do so, but have no problem supporting the national government is that's what will achieve their goals. But Democrats pretty much banished states' rights talk from their vocabularies when they decided not to be the racist party any more, so talk about the virtues of local government and the evils of Washington wind up the more-or-less exclusive domain of Republicans.*
The truth is that when discussing politics, the press and most pols are far more comfortable talking about ideology than about groups, but in fact it's probably much better to think about parties as collections of groups than it is to think of them as proponents of ideas. Granted, what Yglesias calls "historical and rhetorical tropes" can matter; Republicans might, all things be equal, tend to support policies that resonate with the way they talk about politics, but things are rarely equal, and when it comes to a competition between ideology and group interests, odds are very good that group interests will win.
I'd also separate out partisanship from ideology and group-representation as a reason that parties take positions on issues of public policy. In the case of health care, for example, it seems to me that partisanship (the urge to oppose whatever Obama supported because of the belief it would help Republicans candidates in 2010) won out over group representation, which would have led Republicans to fight for a bill that would have helped GOP-affiliated or leaning groups such as doctors, Pharma, and the insurance industry.
All of which, in my opinion, is a perfectly healthy part of how parties facilitate democracy. What's less healthy, to me, is how reluctant many are to acknowledge it. If someone would say that he was supporting the Democrats because, as an African American, he supported the party that works for African Americans -- or that, as a wealthy American, she supports the Republicans because that's the party that looks out for rich folks...well, people don't like to hear that. (And, yes, people have multiple identities, and can choose which of those to identify with politically, so I'm not saying that African Americans "have to" be Democrats, or that Anglo evangelical Christians "have to" be Republicans). I do think it's fine, and regardless of value it's natural in politics, for people to express their preferences with something more universal than "gimme," but in my opinion American political culture is too friendly to ideas and not even close to friendly enough to groups, or political parties.
*And are oddly popular in the abstract among Americans. What I always say to students to balance that out is (1) have you ever been to a DMV?, and (2) James Carville's joke -- I'll race you from Disneyland to Disneyworld, and I get to take the federal roads. Although I should say that Texas DMV offices are fine, and Indiana's, or at least the one that I went to, are terrific. California and Arizona? Not so much, and I can't imagine how bad it must be now in both of those states.