For four years, the leader most capable of unifying the fractious Republican Party has been Barack Obama.

Now the Republicans find their divisions newly revealed in the raw. By exposing the party’s vulnerability to potent demographic shifts, the 2012 results have set the stage for a struggle between those determined to rebrand the Republicans in a softer light and those yearning instead for ideological purity.

But before acceptance comes denial. And the party’s first challenge, it seems in the immediate aftermath, is to find common ground simply in diagnosing the problem. Though some leaders argued that basic mathematics dictates that the party must find new ways to talk about issues like immigration, abortion and same-sex marriage, others attributed Republican losses to poor candidate choice, messaging missteps and President Obama’s superior political operation.

“We continually crank out moderate loser after moderate loser,” said Joshua S. Treviño, a speechwriter in George W. Bush’s administration who now works for the Texas Public Policy Foundation, a conservative group. He said Mitt Romney was part of a “pattern” of Republican nominees, preceded by John McCain, Bob Dole and George H. W. Bush, who were rejected by voters because of “perceived inauthenticity.”