WASHINGTON When Republican leaders in Congress agreed to raise taxes on the wealthy last week, it left the increasingly fractured and feuding party unified on perhaps only one point: that it is at a major crossroads.

From Mitt Romney’s loss on Election Day through the recent tax fight that shattered party discipline in the House of Representatives, Republicans have seen the foundations of their political strategy called into question, stirring a newly urgent debate about how to reshape and redefine their party.

At issue is whether that can be achieved through a shift in tactics and tone, or if it will require a deeper rethinking of the party’s longtime positions on bedrock issues like guns and immigration. President Barack Obama intends to test the willingness of Republicans to bend on those issues in the first months of his new term, when he plans to push for stricter gun control and a comprehensive immigration overhaul.

The coming legislative battles are certain to expose even more division in the party. And with establishment Republicans and tea party activists at times speaking as if they are from different parties altogether, concern is spreading throughout the ranks that things could get worse before they get better.

“The Republican Party can’t stay exactly where it is … and ignore the fact that the country is changing,” said Ralph Reed, the founder of the Faith and Freedom Coalition and onetime leader of the Christian Coalition. “On the other hand, if the party were to retreat on core, pro-family stands and its positions on fiscal responsibility and taxes, it could very quickly find itself without a strong demographic support base.”

Having lost the popular vote in five of the last six presidential elections, Republicans now face a country that is increasingly younger, multiethnic and skeptical of Republican positions on some social issues. The party’s deficit-cutting agenda relies heavily on reducing taxes for the wealthy, which irks middle-class voters, and cutting spending on government programs, like Social Security and Medicare, that are popular with many voters.

Generational change is also robbing the party of some of its most effective political positions. Same-sex marriage, which less than a decade ago was an issue that reliably drove conservative voters to the polls in favor of Republicans, appears to be losing its potency with an electorate increasingly comfortable with gay unions.

Republicans insist that if the party’s disparate factions can come together around a set of economic, social and foreign policy principles in the coming years, they stand a good chance of retaking the presidency and making gains in Congress.

“Republicans will get their mojo back when they define themselves as the party of economic growth and upward mobility,” said Gov. Mitch Daniels of Indiana, a Republican. Daniels said new lawmakers and governors – many of whom are minorities and women – would reshape the party.