Here's a look back at the main characters, events and oddities during the 16-day government shutdown. (Nicki Demarco/The Washington Post)

The GOP establishment has embarked, once again, on a round of soul-searching. But this time, the question is: What will it take to save the Republicans from the self-destructive impulses of the tea party movement?

That the government shutdown was a political disaster for the party that engineered it is widely acknowledged, except by the most ardent tea partyers.

And that near-unanimity presents an opportunity for the establishment to strike back — and maybe regain some control from the insurgent wing.

“You roll them,” advised former Senate majority leader Trent Lott (R-Miss.). “I do think we need stronger leadership, and there’s got to be some pushback on these guys who think they came here with all the solutions.”

Only then, he said, can the party begin to push an agenda and “get things done,” rather than obstruct.

Added another Mississippian, former governor Haley Barbour: “They need to get back on substance.” Barbour noted that the upcoming conference committee on the fiscal 2014 budget presents an opportunity to do that.

Some of the GOP’s leading figures sound as though they have all but given up on Washington.

“Where the party goes from here is to the states,” said Ed Gillespie, a former Republican National Committee chairman. “The party writ large is discouraged with Washington.”

Gillespie is putting most of his energy into the Republican State Leadership Committee, an organization he heads that seeks to elect Republicans to statewide offices. And he noted that some of the party’s biggest stars these days are its governors, who are far removed from the morass in Washington.

The shutdown strategy — to use must-pass bills to fund the government and lift the federal debt ceiling as leverage to gut the new health-care law — never had a chance of succeeding.

And it has left the Republicans with their lowest approval ratings in the history of polling on that topic.

“We’ve got to do a better job of working as a team and having a clear message,” said Sen. Rob Portman (R-Ohio). “I’m optimistic we will after this experience.”

Yet there is no guarantee that the Republicans won’t be back in a similar position as soon as January, when another funding bill will be needed.

“We’ll be looking for any opportunity,” said Rep. Tim Huelskamp (R-Kan.), one of the sponsors of the failed strategy. “We took a shot at it and we fell short, and I think we are waiting around for another battle over Obamacare.”

To avoid a replay of the past few weeks, Republicans must figure out how to deal with several dozen of their most bellicose junior members in the House, and unapologetic figures such as Sen. Ted Cruz (Tex.) who have built a national following and fundraising base on the strength of their obstructionism.

What makes that more difficult, however, is that the tea party movement and some of the groups with which it is aligned have been aggressive about mounting primary challenges to incumbents they deem insufficiently committed to their cause.

Early Thursday, former Alaska governor Sarah Palin posted on her Facebook page: “Be energized. We’re going to shake things up in 2014. Rest well tonight, for soon we must focus on important House and Senate races. Let’s start with Kentucky — which happens to be awfully close to South Carolina, Tennessee, and Mississippi.”

That was a none-too-subtle encouragement of primary challenges to Senate Minority Leader Mitch McConnell (Ky.), as well as Sens. Lindsey O. Graham (S.C.), Lamar Alexander (Tenn.) and Thad Cochran (Miss.).

One complication is that, in many ways, the Republican Party is indebted to its tea party wing, whose passion led to GOP success in the 2010 midterm elections.

But tea-party-fueled primary challenges have backfired in recent election cycles by knocking out Republican candidates who might have won a general election in favor of fringe figures who had no chance.

This time, the establishment has vowed not to let that happen. Crossroads GPS, an independent group co-founded by Gillespie and GOP strategist Karl Rove, has quietly focused more of its resources on what one official described as “thorough candidate vetting.”

That’s another way of referring to opposition research about little-known potential candidates in Republican primaries that the group then shares with other organizations and donors. The idea is to get damaging information in circulation before it is too late to stop unviable contenders from becoming GOP nominees.

Some Republicans lament that it has been difficult to get business leaders and other influential establishment figures to weigh in during primary season

“The problem is, these guys want someone else to do it,” said lobbyist Vin Weber, a former congressman from Minnesota. “They don’t want to get involved in primaries.”

For their part, the tea party forces on Capitol Hill and beyond insist that the political fallout from the shutdown will be temporary.

“There’s been an overreaction to a bunch of this stuff,” said Dan Holler, communications director for Heritage Action for America, the lobbying arm of the Heritage Foundation.

He predicted that the negative attention being directed at the tea party movement and the GOP will fade as the reviews of the health-care law get worse.

“Nobody is surprised that there is division in the Republican Party,” he said.

Holler said that, in the long run, the government shutdown will play well for the GOP as long as the party can articulate that the 16-day stoppage was an attempt to protect Americans from the adverse effects of the Affordable Care Act.

“Now, we’re capable of messing it up,” added Rep. Tom Cole (R-Okla.). “We need to make progress without threatening or without overplaying our hand. If we do that, normal politics resumes, and I think this year, normal politics probably favors us.”

Paul Kane and Jackie Kucinich contributed to this report.