Speaker John Boehner (R-Ohio) generally adheres to the unwritten Republican rule that bars him from allowing votes on bills opposed by a majority of Republicans, even if they would win a majority of the full House.

But he's caved four times this year, allowing big bills to pass with mainly Democratic support. They include repealing the Bush tax cuts for the wealthiest Americans; providing Hurricane Sandy relief; expanding the Violence Against Women act to better cover immigrants, Native Americans, and LGBT survivors of abuse; and this week's bill raising the debt limit and reopening the federal government.

Many presume the Republican House is a black hole sucking President Obama's second-term agenda into oblivion. But the list of Boehner's past retreats offers a glimmer of hope, especially to advocates of immigration reform. Though it has languished in the House, an immigration overhaul passed with bipartisan support in the Senate, and was given a fresh push by Obama in the aftermath of the debt limit deal.

The big mystery that immigration advocates need to figure out: What makes Boehner cave? Is there a common thread? Is there a sequence of buttons you can push that forces Boehner to relent?

Two of this year's caves happened when Boehner was backed up against hard deadlines: The Jan. 1 fiscal cliff and the Oct. 17 debt limit. Failure to concede meant immediate disaster. Reject the bipartisan compromise on rolling back the Bush tax cuts, get blamed for jacking up taxes on every taxpayer. Reject the Senate's three-month suspension of the debt limit, get blamed for sparking a global depression. Boehner held out until the absolute last minute both times, but he was not willing to risk blowing the deadline.

A third involved the response to an emergency: Hurricane Sandy. Conservative groups were determined to block disaster relief because — as with other federal disaster responses — the $51 billion legislative aid package did not include offsetting spending cuts. Lacking Republican votes, Boehner briefly withdrew the bill from consideration, unleashing fury from New York and New Jersey Republicans, including Gov. Chris Christie. While there wasn't a hard deadline to meet, disaster relief was a time-sensitive matter, and the pressure from Christie and his allies was unrelenting. Two weeks after pulling the bill, Boehner put it on the floor, allowing it to pass over the objections of 179 Republicans.

The fourth cave occurred in order to further reform and expand a government program: The Violence Against Women Act. The prior version of the law had been expired for over a year, as conservatives in the House resisted the Senate bill in the run-up to the 2012 election. But after Mitt Romney suffered an 18-point gender gap in his loss to Obama, and after the new Senate passed its version again with a strong bipartisan vote, Boehner was unwilling to resist any longer. Two weeks later, the House passed the Senate bill with 138 Republicans opposed.

Unfortunately for immigration advocates, there is no prospect of widespread pain if reform isn't passed. There is no immediate emergency, nor threat of economic collapse.

But there is a deadline of sorts: The 2014 midterm elections.

If we've learned anything about Boehner this month, it's that he's a party man to the bone. He dragged out the shutdown and debt limit drama for weeks, without gaining a single concession, simply so his most unruly and revolutionary-minded members would believe he fought the good fight and stay in the Republican family. What he won is party unity, at least for the time being.

What Boehner lost for his Republicans is national respectability. Republican Party approval hit a record low in both the most recent NBC/Wall Street Journal poll and Gallup poll.

Here's where immigration advocates have a window of opportunity to appeal to Boehner's party pragmatism. Their pitch: The best way to put this disaster behind them is for Republicans to score a big political victory. You need this.

A year after the Republican brand was so bloodied that the Republican National Committee had to commission a formal "autopsy," party approval is the worst it has ever been. You've wasted a year. Now is the time to do something that some voters will actually like.

There's reason to hope he could be swayed. In each of the four cases in which he allowed Democrats to carry the day, he put the short-term political needs of the Republican Party over the ideological demands of right-wing activists.

Boehner will have to do another round of kabuki. He can't simply swallow the Senate bill in a day. There will have to be a House version that falls short of activists' expectations, followed by tense House-Senate negotiations. Probably like in the most formulaic of movies, and like the fiscal cliff and debt limit deals, there will have to be an "all-is-lost moment" right before we get to the glorious ending. Boehner will need to given the room to do all this again.

But he won't do it without a push. A real good push.