The new conciliatory tone among House Republican leaders this week will soon have to confront an old Washington reality: the GOP’s deep opposition to higher tax rates.

As they begin bipartisan talks on a broad debt deal, White House and congressional leaders face a group of skeptical rank-and-file Republicans in the House who don’t view Tuesday’s Democratic victories as a reason to retreat on taxes.

“The president may think that he’s got a mandate, but so do we. The president may have won a second term, but I won a third term,” Rep. Jason Chaffetz (R-Utah) said Wednesday.

Somewhat chastened by Tuesday’s defeats in the presidential and Senate races, House Republicans said they are willing to work with President Obama on a deal that would include more tax revenue.

But they continue to oppose any deal that would involve higher tax rates.

At the core of the negotiations will be how to increase revenue without raising rates.

Obama and Senate Democrats campaigned on reaching a debt deal that includes raising taxes on the top 2 percent of earners.

But many members of the House GOP caucus have made clear to Speaker John A. Boehner (R-Ohio) that any deal that includes higher individual tax rates would be doomed in the House, where the Republican conference returns with a slimmer but still robust majority next year.

“Raising tax rates is unacceptable. And frankly, it couldn’t even pass the House,” Boehner said Wednesday in an interview on ABC’s “World News With Diane Sawyer.”

Additionally, Republicans demand that any deal — with a target of $4 trillion in savings over 10 years — include steep savings drawn from reforming all entitlements. The Democratic counteroffer is a refusal to discuss any changes to Social Security.

Crafting a package that can gain congressional approval is critical to avoiding the “fiscal cliff” — $700 billion in automatic tax hikes and spending cuts set to kick in after Jan. 1.

The highest hurdle to reaching such a deal may be convincing enough House Republicans, many of whom were swept into power two years ago on the tea party wave, to go along with a compromise that breaks their campaign pledges never to raise taxes.

In secretive talks in July 2011, Obama and Boehner came close to a “grand bargain” that would have included $800 billion in new tax revenue in exchange for entitlement reforms. Some Democrats opposed provisions of that deal, particularly a change in the annual increase in Social Security benefits. But they also had deep concerns about whether there would have been enough GOP votes to support the deal so soon after the 2010 midterm elections.

When Obama demanded more taxes, Boehner backed out of the talks and the issue was never tested among House Republicans — almost all of whom had vowed to never support any tax increase.

A new test may come before New Year’s Day, and in the aftermath of Tuesday’s results the landscape may be somewhat altered.

Few House Republicans have expressed any enthusiasm for the daredevil confrontations of 2011.

That seems partly a result of the election itself, which knocked off two of the most vocal hard-liners — Reps. Allen B. West (R-Fla.) and Joe Walsh (R-Ill.) — and put a scare into a third, Rep. Michele Bachmann (R-Minn.).

“House Republicans believe strongly it would be irresponsible to voluntarily drive off that fiscal cliff,” said Rep. Kevin Brady (R-Tex.), a leader among fiscal conservatives. “I think the dynamics have changed in the sense that the urgency to resolve and address these bigger challenges has risen dramatically.”

But beyond the idea of making a deal, there is little consensus among House Republicans about how to move forward.

Chaffetz said he believes there is enough “common ground” among Republicans and Democrats for a deal that increases revenue by eliminating loopholes and exemptions in the tax code, a simplification process that would likely be worked out through legislation next year.

Sen. Charles E. Schumer (D-N.Y.) agrees that some revenue can be produced from such an approach, but says that without raising rates on the wealthy, it would not be enough to achieve the broad deal that both sides hope to reach.

Some Republicans have voiced support for increased revenue only on the condition that independent scorers such as the Congressional Budget Office could certify that it is the product of economic growth produced by tax reform.

On Wednesday, Schumer called that approach a “Rumpelstiltskin tax fairy tale.”

Some Republican insiders believe that any agreement by Boehner and his lieutenants to increase individual tax rates would amount to political suicide and could lead to a caucus rebellion that would supplant them. At times in 2011, Boehner and House Majority Leader Eric I. Cantor (Va.) seemed to be reading from different scripts, as the speaker pressed hard for a deal with Obama while Cantor protected the conservative flank and opposed many of the proposals.

By all accounts, the duo is now working well together.

But there is still wide agreement that any final deal will require a large number of Democratic votes to pass the House. That’s because there are probably a few dozen Republicans, those from the safest conservative districts, who will not support any deal negotiated with Obama.

Among some of them, the appetite for confrontation is undiminished.

“When I look at the results of the election, it becomes clear to me that the House is now the last line of defense for preserving freedom in this country. The people of South Carolina clearly rejected President Obama’s policies, and I intend to fight on their behalf,” Rep. Jeff Duncan (R-S.C.) said in a written statement.

Duncan said the best solution is for both sides — Republicans in the House and Democrats in the Senate — to first show their best hand: an undiluted vision of how each would solve the problem.

“Republicans have no intention of negotiating with ourselves. If we’re going to solve some of these major issues facing our country, the Senate is first going to have to learn how to pass a bill,” Duncan’s statement said.

“The House passes the most conservative bill possible, the Senate passes their version, then we try to work out the differences. That’s how our Founding Fathers intended for our legislative process to work.”