Congressional Republicans leave Washington for the holidays divided and embittered over the last round of December’s payroll-tax fight, and their lingering unhappiness could complicate negotiations starting in January on a deal for a full-year tax holiday.

Some House Republicans say they feel sold out by their counterparts in the Senate. For their part, Senate Republicans had worried that their House colleagues were harming the GOP’s chances of winning back their chamber by risking a tax increase if House members didn’t get concessions they wanted.

In the House, some rank-and-file House conservatives are deeply disappointed in their own leaders, who in the face of intense political pressure Thursday accepted a two-month deal that House Republicans had almost unanimously rejected just days earlier.

Perhaps no one was more dismayed at the outcome than the nearly 90 freshmen Republicans who came to Washington in January on a tea party wave promising to change the town. Many felt that the year ended with a temporary tax fix that was the epitome of business as usual.

“The House Republicans made a firm, sound point. And when push came to shove, we lost our way,” said Rep. Tim Huelskamp (R-Kan.), a freshman. He said Republicans missed the opportunity to use their new House majority this year to force major entitlement changes, overhaul the tax code and shrink government dramatically.

The tax fix, he said, “was bitterly consistent with what happened all year long.”

Though approved on a bipartisan 89 to 10 vote in the Senate, the 60-day tax deal had been crafted behind closed doors largely by two men: Senate Majority Leader Harry M. Reid (D-Nev.) and Senate Minority Leader Mitch McConnell (R-Ky.).

It was a fallback solution brokered because Reid and McConnell couldn’t agree on how to pay for extending the tax cut for a full year. Twin deadlines were fast approaching: the expiration of the one-year measure that had cut the payroll-tax rate from 6.2 percent to 4.2 percent and the start of lawmakers’ holiday recess.

The temporary deal extended a tax cut many freshmen believe had been embraced by President Obama and Republican leaders merely because it was popular. Opponents argued that it would not stimulate the economy as Obama had maintained. They also said it could harm Social Security funding over time.

“When you start making decisions based on elections, then you run the risk of having the mess we just did,” said Rep. Mo Brooks (R-Ala.)

Congressional Democrats on Friday reveled in their success in forcing Republicans to yield on tax cuts, one of that party’s signature issues.

“I hope this Congress has had a very good learning experience, especially those who are newer to this body,” Reid said after the Senate voted Friday to approve the deal. “Everything we do around here does not have to wind up in a fight.”

Instead, a number of newer members said Friday the message they had gotten was that they must fight even harder in 2012 — and encourage their leaders to stand beside them.

“Here’s my lesson learned: Clearly it demonstrates that common sense doesn’t get in the way of political necessity,” said Rep. Bill Huizenga (R-Mich.). He said the two-month agreement makes no sense for businesses that prepare payrolls on a quarterly basis. “We’re again seeing the lack of an ability to make hard decisions about long-term issues.”

To get a full-year deal on the payroll tax — as well as to extend unemployment benefits and avert cuts in Medicare rates, which are in the same package — Democrats and Republicans will have to bridge a deep divide over whether such items should be funded through cuts in spending or higher taxes on wealthy people.

It’s the same kind of split that bedeviled the 12-member deficit “supercommittee,” which disbanded in failure last month.

Republicans will likely try to eke out concessions from Democrats, knowing that Obama has made the continuation of the tax cut a top priority. In the deal approved Friday, Republicans already got one major win — a requirement that the administration make a speedy decision on whether to allow construction of the Keystone XL oil pipeline.

But their bargaining hand will not be strong, because a deal is now more important for the GOP. That’s because party leaders have spent the past week insisting that a full-year cut is necessary for the economy. And they have gotten a taste of the political consequences of letting Obama portray them as willing to let taxes rise for 160 million workers, as he has in recent days.

The $33 billion package was approved Friday in voice votes in the House and Senate, each lasting only a couple of minutes, and signed into law immediately by Obama.

In brief remarks after the bill’s passage Friday, Obama praised Congress for ensuring that Americans’ payroll taxes will not rise next month. And he said lawmakers should move quickly in January to extend the tax cut for a full year.

“When Congress returns, I urge them to keep working, without drama, without delay, to reach an agreement that extends this tax cut, as well as unemployment insurance, through all of 2012,” he said. “It’s the right thing to do because more money spent by more Americans means more businesses hiring more workers. That’s a boost for everybody, and it’s a boost we very much need right now.”

“Aloha,” he concluded, departing the White House to join his family in Hawaii for a Christmas vacation he had delayed because of the tax fight.

House Speaker John A. Boehner (R-Ohio), who stood alone at the microphones Thursday night to announce that he would accept the two-month deal in exchange for a promise from Reid to immediately begin negotiations over a full-year deal in January, presided over the brief House session.

And then he left the Capitol, without offering further comment.

This month’s debate revealed a deep division among Republicans about whether the payroll-tax cut has been a good idea. Those voices are likely to grow stronger in January because of unhappiness with how leaders handled the fight this month. In particular, many House Republicans say they feel betrayed by colleagues in the Senate.

As the week wore on, a steady stream of Republican senators came forward to say the House should abandon its demand for further negotiations to get a full-year deal that might include elements such as a continued pay freeze for federal employees.

Boehner’s hand was ultimately forced by McConnell, who after days of silence emerged Thursday to urge the House to back the 60-day fix in exchange for Reid appointing negotiators to start new talks in January.

“I feel really let down by the Senate Republicans,” said Rep. Jason Chaffetz (R-Utah). “We were under the impression that we were strengthening the Senate’s hands and that by passing this tough bill it would give Mitch McConnell more room to negotiate.”

Instead, he said, McConnell “just rolled over to get his belly itched.”

Staff writers David A. Fahrenthold and Felicia Sonmez contributed to this report.