After their 2010 triumphs, House Republicans had once expected 2012 to bring even more victories and vindication of the path they chose over the past two years. They controlled the agenda in Washington for much of that time, putting President Obama on the defensive and severely hurting his popularity.

But in the process, they suffered some self-inflicted damage. Although they retain a robust majority and control of the House, the GOP brand was mostly routed Tuesday when it traveled beyond the boundaries of gerrymandered districts meant to protect the party’s vulnerable incumbents.

As a result, Wednesday began the unveiling of a more conciliatory House GOP pledging compromise and cooperation in the face of an altered Washington landscape that is suddenly more Democratic. Gone was the harsh rhetoric of the 2011 fiscal showdowns, and in its place was a declared willingness to consider raising tax revenue as part of a larger debt deal that would include entitlement reform and deficit reduction.

Tuesday’s results showed poor performances by GOP candidates with Latino voters up and down the ballot, and some Republicans on Wednesday signaled a willingness to strike a comprehensive immigration deal, even if it sparks anger among conservative activists.

More broadly, House GOP leaders hinted that the days of partisan brinksmanship may be over. “We’re ready to be led, not as Democrats or Republicans, but as Americans,” House Speaker John A. Boehner (R-Ohio) told reporters, suggesting that he would work with Obama on fiscal issues immediately and others in the long term. “We want you to lead — not as a liberal or a conservative, but as the president of the United States of America. We want you to succeed.”

That tone is a long way from remarks two years ago by Senate Minority Leader Mitch McConnell (R-Ky.), who called Obama’s reelection defeat his top priority.

The question in the next Congress, as it was in the previous, is whether Boehner has enough muscle or capital to muster the requisite votes to make him a reliable negotiator.

The speaker held a conference call Wednesday afternoon with GOP lawmakers to brief them about the election and how they would approach the months ahead. Some Republicans embraced a new direction, seeing a demographic shift that has left the House GOP isolated and shrinking while Democrats picked up Senate races and retained the White House.

“Republicans got worked. Republicans got worked,” Rep. Cory Gardner (R-Colo.), who won a second term Tuesday, said in a telephone interview. He warned that the party could turn into “the Lions Club: going to the same meeting every week, with fewer and fewer people.”

Some others read the situation differently, seeing a new imperative for House Republicans to stand strong against Obama’s agenda in the face of a Mitt Romney defeat.

“My candidate just got waxed. And I understand that there will be people saying, ‘The message is, we should raise taxes.’ Even if that’s the mandate . . . that’s not an option that’s on the table for us,” said Rep. Mick Mulvaney (R-S.C.), who is also a member of the influential Class of 2010.

On Wednesday, Senate Majority Leader Harry M. Reid (D-Nev.) said he was willing to work on legislation with House leaders, under certain conditions. “I want to work together, but I want everyone to also understand you can’t push us around,” Reid told reporters.

Within three months of taking the majority, House Republicans forced Obama into $38 billion in 2012 spending cuts that he did not want, threatening to shutter the federal government if he did not relent. By the summer of 2011, House Republicans refused to agree to lift the federal borrowing limit by $2 trillion unless there were corresponding cuts of an equal amount over the next decade.

An eleventh-hour deal averted a potential debt default, but the moment caused a crisis in public confidence in the federal government that still echoes. Congressional approval ratings sank to about 10 percent, an all-time low, and Obama’s image sank, as well.

Democrats readied a campaign to paint the “tea party Congress” as the key obstacle to a bipartisan economic recovery, but in House races, the effort largely fizzled.

Boehner will return with a majority of at least 235 seats, a loss of about a half-dozen GOP seats. The relatively small number of losses was mostly the result of the decennial redistricting process, during which the GOP worked to shore up the most vulnerable Republicans. Of the more than 80 freshmen Republicans who sought reelection, only 10 have officially lost their seats.

However, in a sign of how poorly House Republicans performed beyond those carefully drawn districts, five were nominated to the Senate races, and four of them lost, despite beginning as favorites. (Meanwhile, four out of five House Democrats won their Senate races.)

Moreover, the House Republican ideological leader, Rep. Paul Ryan did little to help Romney in his home state of Wisconsin, which Obama won handily. Ryan won reelection to his own southern Wisconsin district, but with less than 55 percent of the vote, his worst showing ever.

Gardner, whose eastern Colorado district includes large numbers of Latinos, said the hard-line positions of some Republicans on immigration issues creates a “fear factor” that drowns out other GOP messages. “We talk about family values, we talk about hard work, small businesses,” he said. “But, hey, it doesn’t matter.”

Now, Gardner said, House Republicans should seek to work on immigration issues with Obama and search for a more diverse crop of candidates. “A big tent,” he said, “is no good if it’s empty.”

Rep. Adam Kinzinger (R-Ill.), another freshman who won reelection, said it was important not to be seen as obstructionist. “Govern from where we are philosophically. But govern,” he said. “Divided government isn’t always bad, unless it’s too divided. . . . Let’s make a divided government that works.”

Rep. James Lankford (R-Okla.), one of the more conservative members of the House freshman class, said he believed the election showed Republicans had lost a messaging war on their top issue: the debt and federal spending.

“The Democrats did a good job at painting us as people who just want to slash and burn and cut. We failed to say how it’s important to balance your budget,” Lankford said. He said that Republicans in Congress may have become too focused on the vast numbers involved, and not on the ground-level impacts that would hit home with voters. “We’ve got to do a better job of explaining how this connects to actual families.”