But since Election Day, he has seemed to be choosing between them more than in the past. His decision after the Newtown massacre to embark on a full-scale effort to crack down on gun violence showed him to be less shackled to political wisdom about what is possible or electorally wise. His willingness to stare down Republicans over raising the debt limit — and winning — showed that he is less likely nowadays to start a negotiation by moving to the center and trying to find common ground.

To some Republicans, it is what they warned of all along: a president who ran as a centrist proving to be an unreconstructed liberal. It was no doubt hard for some of them to accept a scolding for treating “name calling as reasoned debate” — a phrase in his Monday address — from a man who won re-election by excoriating Mitt Romney as a job-killing plutocrat.

“I think all Americans would hope that President Obama, now that he’s not facing re-election, would actually sit down and honestly work with Republicans who are very sincere in our desire to fix these problems,” said Senator Ron Johnson, Republican of Wisconsin.

But, Mr. Johnson added, that was not the sentiment he detected from Mr. Obama on Monday. “You’ve got to sit down in good faith,” he said. “But I just don’t see that with this president.”

Representative Pete Sessions, Republican of Texas, said, “I’m surprised we’ve so abruptly noticed after this election we’re now managing America’s demise, not America’s great future.”