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Republican lawmakers are so fearful of social issues, in fact, that House leaders ignored intense objections from conservatives last week and allowed the passage of Democratic legislation on domestic and sexual violence against women.

All of which helps explain why Speaker John A. Boehner and Congressional Republicans have been so intent on facing down Mr. Obama in their budget dispute, taking on the economic and political risks associated with the automatic spending cuts that went into effect on Friday to make the point that they would not be maneuvered by the White House into accepting more tax increases.

"I'm going to say it one more time," Mr. Boehner said Sunday on "Meet the Press" on NBC. "The president got his tax hikes on January the first. The issue here is spending. Spending is out of control."

Four months after Mr. Obama won a second term, the only issue that truly unites Republicans is a commitment to shrinking the federal government through spending cuts, low taxes and less regulation. To have compromised again and agreed to further tax increases or to roll back spending cuts would have left Republicans deeply split and, many of them say, at risk of losing the core of the party's identity.

"If the voters can't rely on us to stand up to the runaway train of entitlements and deficits and federal debt, what can they count on us for?" said David Kochel, a Republican consultant in Iowa. "We're going to have disagreements on other issues. This is one we have to agree on."

One of the most striking characteristics about the political climate in the months since Mr. Obama's re-election is that on issue after issue, it is no longer entirely clear what it means to be a Republican, with the party more divided than ever on domestic policy and a debate breaking out over how best to invigorate the conservative movement.

In that sense, the budget showdown is not just about cutting $85 billion out of government spending over the next seven months. It is, to many conservatives, about showing that Republicans still have the will, the leadership and the public support to use this moment to halt, or at least slow, an ideological pendulum swing from the right, where the nation seemed to be heading after the 2010 midterm elections, to the left, after the 2012 campaign.

"The sequester and winning that fight — however you define what winning means — is critical for the party," said Ralph Reed, the longtime social conservative leader.

There are risks for Republicans in taking a hard line on the spending cuts, especially if the unemployment rate jumps and the economy slows. Democrats are highlighting estimates by the Congressional Budget Office that the fiscal cutbacks could leave the economy with 750,000 fewer jobs this year than it would have otherwise. And they are warning that while the effects of the budget cuts will play out slowly, the cuts will eventually hit voters in noticeable ways.

"I don't see how 'I'm fine with the sequester' differs from 'I'm fine with slower growth and continued high unemployment' or longer delays at airports, fewer Head Start slots, furloughs among civilian DOD folks and so on," said Jared Bernstein, a former economic adviser to Vice President Joseph R. Biden Jr. and a senior fellow at the Center on Budget and Policy Priorities, a liberal research group. As the cuts become real, he said, they will "deeply discredit" those who foster the idea that government spending is inherently wasteful and that there is no tangible cost to reducing it.

But to the degree that Mr. Boehner has prevailed in the first round of the current fight — the skirmishes over the across-the-board cuts and the longer-term questions about addressing the national debt will go on for months and years — it could not have come at a better time for the morale of Republicans.

In the view of some conservative commentators, the Republican Party and the conservative movement are at one of their lowest points in years — "leaderless and nearly issueless," as Rich Lowry, the editor of National Review, wrote last week in an opinion piece for Politico that set off debate in the party.

With the party divided on so many other fronts, feelings are running especially high when it comes to holding the line on taxes. Conservatives are excoriating Gov. Bob McDonnell of Virginia, a Republican, for championing a tax increase to pay for transportation projects.

Chris Chocola, the president of the Club for Growth, a group that targets Republicans it considers to be insufficiently conservative on fiscal policy, said, "Republicans have not had a hard time convincing the base they are conservative enough on social issues. They have had a hard time convincing the base they are sincere on economic issues because they have grown the government."

The internal divisions after the party's fourth loss in the last six presidential elections and the recognition that inexorable demographic shifts are working against a reliance on its traditional base have set off the most fundamental debate since the Reagan years about the future of conservatism.

Liberals dismiss the exercise as a sham intended to distract attention from enduring ties between wealthy interests and conservative policy and politicians. That is a point implicitly recognized by many Republicans, who have put at the heart of the debate about their future the question of how to defend the party's small-government philosophy without being portrayed by Democrats as imprudently abstemious and against middle-class interests.

They are beginning to talk about focusing less on reductions in marginal income-tax rates — long the primary goal of Reagan-era supply-siders — and more on tax changes, like increasing the child credit or reducing payroll taxes that would more directly address the stagnation in middle-class incomes and growing inequality.

Portraying deficit reduction as an end unto itself, Mr. Reed said, would miss an opportunity for Republicans to connect with voters who are concerned about their families and neighborhoods.

"There's a danger if the Republican Party is only seen as the party that cuts government spending, because that is a necessary but insufficient rationale for a majority party," he said.

In an article in Commentary entitled "How to Save the Republican Party," two veterans of President George W. Bush's White House, Michael Gerson and Peter Wehner, said the party could rebuild itself with a social mobility agenda that extends beyond low taxes to cut back tax breaks for corporations, break up big banks, help poor people reach and graduate from college, and improve health care for children.

"Rather than being exclusively focused on budget numbers or individual economic rights, Republicans would be demonstrating a limited but active role for government: helping individuals attain the skills and values — the social capital — that allow them to succeed in a free economy," they wrote.

For now, the debate about the future of the party remains detached from legislative politics. Mr. Boehner has more than enough trouble navigating the demands of various stripes of conservatives on a day-to-day basis without introducing a more activist strain of policy into the Republican agenda. In talking about taxes, he continues to stress reducing income-tax rates.

But with the automatic budget cuts now probable to remain in force for some time, if not permanently, attention in Washington will shift to long-term budget plans to be introduced in the coming weeks by Representative Paul D. Ryan of Wisconsin, the Republican vice-presidential nominee last year, Senate Democrats and Mr. Obama.

It is those documents that provide both sides with their best opportunity to define for voters what they stand for over the remainder of Mr. Obama's presidency and beyond.