The fallout will help determine whether the GOP might return to heights approximating the Ronald Reagan years or, as some fear, suffer even deeper losses as the nation’s Democratic-leaning Hispanics increase in number.

“The party is clearly in some sort of identity crisis,” said Rick Tyler, a past aide to former House Speaker Newt Gingrich.

Hard-core conservatives, furious at President Barack Obama’s re-election in the face of a weak economy, called for a wholesale shift to resolutely right positions on social and fiscal matters. Some demanded that party leaders resign.

Establishment Republicans largely shrugged off the tirades. But they split into two main camps themselves, portending potentially lengthy soul-searching, especially in Congress.

One group calls for calm and a steady course. It emphasizes that the party still controls the House, and notes that Obama’s popular-vote margin was smaller than in 2008.

“The Republican Party is exactly right on the issues,” said Terry Holt, a veteran GOP strategist with close ties to House Speaker John Boehner, R-Ohio. The party mainly needs to nominate candidates who can relate to average Americans better than multimillionaire Mitt Romney did, Holt said.

Some other Republicans, however, see bigger problems. The party must shed its “absolutism on issues like tax increases,” which congressional lawmakers oppose at virtually every level, said John Ullyot, a former Republican Senate aide.

“The only way the party is going to move more to the middle is when we get sick of losing,” he said.

That’s essentially what Democrats did in the 1990s. Demoralized after big losses by presidential nominees Walter Mondale and Michael Dukakis — and still mindful of George McGovern’s 1972 disaster — Democrats turned to a centrist Arkansas governor, Bill Clinton. He won two elections, repositioned the party and served as Obama’s top surrogate this fall.

Some activists in both parties say Republicans eventually must follow suit to survive. But their primaries are dominated by staunch opponents of tax hikes, abortion, immigration reform and government regulations. Until and unless that changes, a shift toward the center may be impossible.

“It’s harder for the Republicans, because they are more ideological than Democrats,” said Democratic strategist Doug Hattaway. “The religious fervor of the Republican base makes it hard to change or compromise, even though that’s what’s needed to remain viable as a party.”

While Holt and others say the Republican Party is aligned with most Americans on big issues, Tuesday’s exit polls raise doubts in some areas. Six in 10 voters said abortion should be legal in all or most cases, the highest share saying so since the mid-1990s. Two-thirds of voters said illegal immigrants working in the United States should be offered a chance to apply for legal status.

Nearly half of all voters supported Obama’s plan to raise taxes on couples’ incomes above $250,000. Thirteen percent said taxes should be increased on all Americans, and 35 percent said no one should pay higher taxes.

Boehner and Senate GOP Leader Mitch McConnell will stand at the center of the intra-party debate. Within days they must decide how to negotiate with Obama and Democratic lawmakers on the looming “fiscal cliff,” a package of major tax hikes and spending cuts scheduled for the new year.

McConnell issued a defiant statement Wednesday. “The voters have not endorsed the failures or excesses of the president’s first term,” he said. “They have simply given him more time to finish the job they asked him to do together with a Congress that restored balance to Washington.”

Boehner was more conciliatory in tone when he addressed reporters Wednesday. But he recommended Romney’s tax package — including rate cuts for everyone and the elimination of yet-to-be-named deductions — which he said would create a net increase in government revenue.

Obama has insisted that the wealthiest Americans pay higher tax rates, as they did under Clinton. Many Democrats in Congress agree.

Republican insiders, meanwhile, nervously focused on an approaching problem that could produce even bigger presidential losses in future years. The GOP relies overwhelmingly on white voters, a steadily shrinking share of the population. Hispanics, the nation’s fastest-growing group, have bristled at Republican attacks on illegal immigration, which some people consider a slap at all Latinos, legal or not.

Republican campaign pros said the party must find a way to temper the talk about immigration without infuriating conservatives who oppose “amnesty” for those who entered the country illegally.

“You can’t just say ‘If you fix the tone, you fix the problem,’” said Republican consultant Terry Nelson. “We have to figure out what kind of policy solutions we have for this.”

Ullyot said congressional Republicans should embrace more lenient immigration policies immediately.

On still another front, many Republicans said their party must find ways to appeal to women, who voted heavily for Obama. The party cannot give people the impression that opposing abortion is its top women-related issue, said Rep. Ileana Ros-Lehtinen, R-Fla.

“Single moms are part of our American fabric,” she said. “Let’s not keep thinking that the American family is made up of a mom and a dad and two kids and a picket fence and a dog and a cat. It’s made up of a lot of single moms struggling to make ends meet. … We need to get a program to say ‘we care about you.’”