Reports that House Speaker John Boehner is searching for a face-saving exit from the stalemate on Capitol Hill suggests a Democratic dream come true, that the shutdown is cratering the Republican Party. A sampling of five major polls last week surveying public attitudes toward the shutdown find “a Republican Party deeply at odds with the electorate,” says Democratic pollster Geoffrey Garin. “Only 20 percent of the electorate thinks this is a rational or reasonable approach.”

Four in 10 Republicans say it’s more important to fund the government than defund Obamacare. By making the standoff a debate about them and their approach rather than the health-care law, “Republicans completely squandered whatever political advantage they may have had on this,” says Garin, “and they are really at the tipping point of making 2014 a referendum on their extreme and irresponsible tactics.”

The fight is far from over, and Republicans may have found a sweet spot in proposing a series of bills to restore parts of the government, putting the Democratic Senate on the defensive. But absent any clear strategy by the GOP, Garin believes the impasse has the potential to turn next year’s midterm elections into a referendum on Republican management of the House, as opposed to the failure of Obamacare, as the GOP had hoped.

The data in Garin’s polling suggest that the only way the Republicans get out of the hole they dug for themselves, and avoid a deeper hole with the looming debt-ceiling deadline, is to delink Obamacare. “Republicans are in the process of figuring out they’re in danger of coming away from all this with nothing,” says William Galston, a senior fellow at the Brookings Institution and a veteran of the 1995–96 shutdown when he was President Clinton’s domestic-policy adviser. Obama continues to demand “a clean CR," a continuing resolution to fund the government with no strings attached, but he might settle for a slightly soiled one, says Galston, one that opens the government without condition but gives the GOP something Boehner can call a victory. Repealing an Obamacare tax on medical devices has bipartisan support, and a formula for replacing the lost funds has reportedly been identified, but leaders in both parties rejected it.

Republican Rep. Marlin Stutzman of Indiana was widely mocked for saying the shutdown is about more than Obamacare, it’s about Republicans being “disrespected.” “We have to get something out of this. And I don’t know what that even is,” he said. He later walked back his remarks, but he spoke the truth. Boehner is trying to figure out what he can get, and what members of his caucus will accept without staging a revolt.

The 1995–96 shut down ended with “total capitulation” by House Republicans led by Speaker Newt Gingrich, says Galston, who recalls “the pressure building on Republicans every day—it was Chinese classic water torture, drip, drip, drip, and it finally wore a hole in the rock.” On the third day of the shutdown, Gingrich confessed to reporters that he was motivated in part by Clinton’s “snub” of himself and Senate leader Bob Dole on a long flight to Israel to attend the funeral of assassinated Prime Minister Yitzhak Rabin.

“You’ve been on the plane for 25 hours and nobody has talked to you, and they ask you to get off the plane by the back ramp,” Gingrich fumed. “You just wonder, where is their sense of manners? Where is their sense of courtesy?” The White House expressed bewilderment that Gingrich saw the flight as a negotiating opportunity, pointing out that Clinton, grieving over the loss of his friend and partner in the peace process, was closeted in the front of the plane with former Presidents Carter and George H.W. Bush. The Daily News front page screamed, “CRY BABY,” and featured a drawing of Gingrich in diapers waving a bottle and pitching a tantrum.

“The Republicans are doing the only thing that could conceivably endanger their majority.”

The fight then was over spending priorities. That spring Clinton had overridden the advice of most of his advisors to endorse a balanced budget. “Once we do this,” he told them, “the debate will shift from whether to balance the budget to how to balance the budget,” terrain friendlier to Democrats. Clinton vowed to protect Medicare, Medicaid, education, and the environment, and the debate with Gingrich turned on whose values did the voters support. “And that narrative line was drummed into the heads of the voters,” says Galston. “It wasn’t just Newt the Cry Baby.”

Obama has laid some of the same groundwork in his advocacy for “investments to build the middle class,” but he’s facing a tougher crowd, and Boehner isn’t as easy to caricature as Gingrich, an in-your-face leader as opposed to Boehner, who’s more like a dead man walking through this crisis.

Before the shutdown, Democrats were pretty much reconciled to falling short in efforts to retake the House next year. With so many gerrymandered seats, the 17 pickups Democrats need seem out of reach and may still be. But Galston points to research that says Democrats can overcome the structural disadvantage they face if they add 5 percentage points in their overall votes for the House, then they would have a better than 50 percent chance of winning control. “The Republicans are doing the only thing that could conceivably endanger their majority,” Galston says. The latest Quinnipiac and Pew polls show Democrats with a 9-point generic edge, twice what it was in late August. “After the dust settles next November and the generic Democratic edge is 9 points, I guarantee you the next speaker is Nancy Pelosi,” he says.

On the Senate side, Republicans need six seats to gain control of the chamber. In just about every contested race, Republicans are battling each other in primaries to see who can be the biggest cheerleader of the government shutdown. “If Ted Cruz becomes the public face for the Republican Party, it will be a cold day in hell before they have a Senate majority,” says Galston. The last shutdown 17 years ago helped reelect Clinton. This one could give Obama the Congress he wants.