So what really happens when Republicans tackle immigration?
Many Republican voters are not opposed to reforms, at least in principle. A February CBS News poll showed a majority of Republicans would either allow currently illegal immigrants living in the U.S. to stay and eventually apply for citizenship, or be allowed to stay legally in some manner (without citizenship.) About four in 10 Republicans do say those residents should be required to leave.
Conservatives within the Republican party actually aren't much different, despite conventional wisdom that sometimes assumes they're all adamantly opposed.
Most self-identified conservative Republicans do say they support some form of legalization when asked, either by permitting a chance to apply for citizenship (34 percent) or allowing temporary residency (23 percent) - which is more or less the same breakdown we see among Republicans, overall. These numbers ought not entirely surprise given that conservatives make up the bulk of the party, but the chart helps illustrate the broader point.
This gets obscured sometimes, perhaps, because compared to Democrats and independents, Republicans are the partisan group least likely to back any forms of legalization - but that's only relative to independents and Democrats.
Immigration and the Republican Party invariably link up to discussions of 2016, too - as we saw with the Jeb Bush discussion - usually framed as a supposed conundrum for presidential candidates (and many in the party's leadership) who see the issue as a starting point for reconnecting with Hispanic voters, but first need to get through the primaries where - the theory goes - the base would punish them for having the wrong stance.
But in recent years, we've haven't seen clear data that it works that way. Back in 2007, the year President George W. Bush's comprehensive immigration reform push was thwarted - one that contained a guest worker program - many Republicans said they were also okay with it in the polls, and that December, heading into the 2008 nomination contests, 56 percent of would-be Republican primary voters accepted some form of legalization either through a guest work program or citizenship.
Then in 2012, immigration didn't appear to have been a deal breaker in the primaries, to the extent we could measure it, though the narrative often tempted us to think so. Some felt Texas Gov. Rick Perry's chances were damaged by comments about education and immigrant children; that Newt Gingrich's reticence to deport long-standing community members cost him support.
Maybe so, but just because those candidates lost doesn't mean immigration was the main reason, as it's hard to isolate those comments from other troubles in those campaigns. Perry had other stumbles in the debates, Gingrich was often outgunned in advertising. In the Florida primary, exit polls told us that Gingrich trailed Mitt Romney among voters of all views on immigration. In Arizona - which wasn't much of contest, but is a state where the issue is central - the Romney vs. Rick Santorum difference was the also the same no matter voters' policy preference. (And a side note, recent polls, including our own taken before this controversy, have had Jeb Bush among the top names Republicans want to see run next year.)
Admittedly, this is a topic where the rhetoric of the debate can change easily, which means the past isn't a perfect indicator. Much of the hesitancy comes from worries about amnesty, specifically, and definitions of what constitutes amnesty in a policy could shift in the course of a more robust campaign, and maybe the responses to future polls along with them. Those unknowns make for a more uncertain political environment for any candidate or legislator.
Moreover, although we reported last year that many conservatives felt political opposition to a reform package might be overstated, we're now entering an election year where Republicans can very reasonably envision both a House and Senate victory. So there may be less political will to try anything that even remotely risks angering core voters.
But whether or not a bill happens in the House, this looks to remain a hot topic for the 2016 field -- and crowded field of candidates always looks for spots and issues to differentiate themselves. This is certainly one where there's room to do that, as there is no single prevailing view within the party's rank and file.