WASHINGTON – 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.

But it was a major year-end political victory for President Obama, a big slice of humble pie for House Republicans and a blow to House Speaker John Boehner, R-Ohio, who will have an angry band of tea party lawmakers to deal with when Congress returns to Washington next month.

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President Obama

Back-to-back voice vote approvals of the two-month special measure by the Senate and House came in mere seconds with no debate, just days after House GOP leaders had insisted that reopening negotiations on a full-year bill was the only way to persuade them to prevent a tax increase on Jan. 1.

Obama immediately signed the bill into law.

“I said it was critical for Congress not to go home without preventing a tax increase on 160 million working Americans, and I’m pleased to say that they got it done,” a buoyant-looking Obama said at the White House before dashing off for his delayed holiday vacation to his home state of Hawaii.

Meanwhile, some House Republicans say they feel sold out by their counterparts in the Senate. For their part, Senate Republicans had worried 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.

Some rank-and-file House conservatives are deeply disappointed in their own leaders, who caved to intense political pressure Thursday and 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 freshman 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 89-10 vote in the Senate, the 60-day tax deal had been crafted behind closed doors largely by just two men: Senate Majority Leader Harry 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 the president 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 Huizengam, 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 decision 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.

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.

Boehner, who had stood alone at the microphones Thursday night to announce 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.

He left the Capitol without offering further comment.

— The Associated Press contributed to this report.