As ugly as they were, the “fiscal cliff” negotiations produced something Washington hadn’t seen in a long time: strongly bipartisan votes in the House and the Senate on a big, contentious issue.

The question now is whether that victory of pragmatism over ideology offers a new model of governing as President Obama approaches his second term and the shattered Republican Party tries to regroup.

The answer: probably not, though it may have helped define the terms of engagement for the battles to come.

“We’re still in a pretty fluid period as to how and whether the two sides can work together,” Rep. Tom Cole (R-Okla.) said Wednesday.

White House officials see the deal brokered this week as a critical short-term victory with long-term consequences — forcing House Republicans to surrender on one of their party’s core tenets. The agreement, approved with comfortable bipartisan majorities on New Year’s Day, raised taxes on net income over $400,000 while holding rates steady for income below that amount.

The impact of tax increases on taxpayers

From the outset, the Obama administration approached the negotiations as an opportunity, in the words of one top official, to “readjust the balance” from 2011, when the president became mired in negotiations to lift the debt ceiling and failed to produce a larger deal that would have trimmed $4 trillion over a decade.

With Obama’s reelection, White House strategists were determined that Republicans recognize that a new political dynamic had taken hold, one that could cement the president’s legacy with legislative victories on issues including immigration.

“A year ago, the idea that the Republican Party would agree to raise a dollar in revenue from the wealthy was seen as a pipe dream,” said Dan Pfeiffer, the White House communications director. “The Republicans wouldn’t agree in the room, so the president took the battle to the country and made it a centerpiece of his campaign and forced the Republicans to do the one thing they said they would never do. This isn’t the best way to govern, but it was the only option available.”

The vote marked the first time in more than a decade that a measure passed a GOP-controlled House despite being opposed by a majority of Republicans in that chamber. So splintered was the party that even its leadership was divided: Speaker John A. Boehner (Ohio) supported the deal, while his second- and third-in-command, Majority Leader Eric Cantor (Va.) and Majority Whip Kevin McCarthy (Calif.), opposed it.

The experience was especially disillusioning for those hard-line Republicans who were elected in the tea party era.

“I know a lot of folks walked out with a real bad taste in their mouths,” said Rep. Tim Huels­kamp (Kan.), a freshman who has emerged as a leading critic of Boehner’s since the speaker removed him from a plum committee assignment last month. “It wasn’t Republicans clapping and cheering. It was Democrats.”

Republicans insist that this will not become a pattern in the House, where governance by “a majority of the majority” has been the watchword since the late 1990s.

It “was an anomaly,” said Cole, a stalwart conservative who had argued internally that the caucus should allow tax rates to rise for the wealthy to pave the way for a broader spending fight in 2013.

He predicted that a “more natural pattern” will return for future deficit-reduction debates, “and that’s a majority of the Republican conference with a minority of the Democratic conference.”

But Democrats say little will get done if that is the case.

“The only way forward on some of these big fiscal issues will be if the speaker allows the people’s house to work its will,” said Rep. Chris Van Hollen (Md.), the senior Democrat on the Budget Committee. “You cannot govern with this artificial rule that you will only allow a vote in the House if a majority of Republicans support it.”

While the fiscal-cliff showdown is over, it will be followed by others in the next few months: over lifting the nation’s $16.4 trillion borrowing authority, over $110 billion in automatic spending cuts and over a funding bill to keep the government operating after a stopgap measure expires in March.

The debt ceiling in particular is something that Obama had hoped to resolve in the cliff negotiations. White House officials insist, however, that he will stand firm when the deadline arrives and will not allow Republicans to use that must-pass legislation — necessary to avoid a government default — as an opportunity to open broader talks over spending and entitlement reform.

“I will not have another debate with this Congress over whether or not they should pay the bills that they’ve already racked up through the laws that they passed,” Obama said after the approval of the cliff deal.

Boehner spokesman Michael Steel dismissed that kind of talk: “Wishful thinking is not a plan.”

Republicans note that the next battles will be fought over spending. And on that issue, they say, public opinion is on their side — so, unlike in the fiscal cliff negotiations, they think they will have the upper hand politically.

“We’ve done some good for the country. We’ve taken care of the revenue side of this debate,” said Senate Minority Leader Mitch McConnell (R-Ky.), who stepped in and brokered the deal after Boehner stumbled. “Now it’s time to get serious about reducing Washington’s out-of-control spending.”

“It’s the debate we’ll have next,” McConnell added. “And it’s a debate Republicans are ready for.”

Obama’s team has also drawn criticism from its side. Many liberals, who have long considered the president a weak negotiator, believe he gave too much ground by abandoning his campaign pledge to raise taxes on all households earning more than $250,000 a year. They also worry that he won’t have as much leverage as he seems to think he will if Republicans try to use the debt-ceiling deadline to force deep cuts in entitlements.

Sen. Tom Harkin (D-Iowa), who voted against the tax deal, called it “shortsighted to look at these issues in isolation from one another, especially when congressional Republicans have been crystal-clear that they intend to seek spending cuts to programs like Social Security just two months from now, using the debt limit as leverage.”

Yet some White House allies say the fiscal cliff experience provides a strategy for future policy fights, even beyond the budget battles to come over the next 90 days. The administration is planning to take an aggressive stance, for instance, in pushing for a path to citizenship for millions of illegal immigrants.

Boehner and other Republicans, citing their party’s dismal performance among Hispanic voters in 2012, have said they are willing to consider immigration legislation this year. But many conservatives are skeptical of a broad measure, and White House officials have told immigrant advocacy groups to prepare for a public push similar to the campaign-style events that Obama held in connection with the fiscal cliff debate in recent weeks.

Meanwhile, White House aides are debating whether they should take the unusual step of drafting an immigration bill or instead lay out principles that could serve as a rallying point. Pro-immigration Republicans will be recruited to help, among them evangelical pastors and small-business owners.

Said one outside strategist who is familiar with White House thinking but who discussed the strategy on the condition of anonymity: “The second term rests on the hypothesis that the House Republicans can be broken.”

Rosalind S. Helderman contributed to this report.