HEMPSTEAD, N.Y. — President Barack Obama and Republican nominee Mitt Romney fought to a draw Tuesday in their second debate, but it was bloody, with each candidate scoring points in several heated exchanges.

From jobs to gas prices to immigration to taxes, Obama and Romney tangled aggressively, sometimes talking over each other and the moderator, CNN’s Candy Crowley. Snap polls and focus groups offered the campaigns and their supporters something to feel good about, although each team and their surrogates insisted that they dominated the town hall-style event.

“This was in New York — this was a New York-style debate,” Rep. Peter King (R-N.Y.) told reporters on the campus of Hofstra University on Long Island. “I disagree with a lot of what President Obama said tonight, but I don’t disagree with the two of them going at. That’s what it’s about.”

Perhaps the defining moment of the debate — or at least the one that was most ready-made for replay on local television, talk radio and cable news — came in an exchange about the attacks that killed American diplomatic personnel in Benghazi, Libya.

After the president spoke about the attack, Romney jumped in.

“I want to make sure we get that for the record because it took the president 14 days before he called the attack in Benghazi an act of terror,” the governor said.

“Get the transcript,” an angry-sounding president replied.

Crowley cut in and said that Obama did, in fact, call the attack an “act of terror.”

Obama, seated in his chair, pointed his finger at the moderator’s dais and said, “Can you say that a little louder, Candy?”

The audience began to applaud.

As TV fact-checkers pointed out after the debate, Obama’s quote from his Rose Garden speech the day after the attacks was “No acts of terror will ever shake the resolve of this great nation.” But whatever the nuances of fact, the moment, both to viewers at home and pundits on TV, looked like a win for a president who badly needed one.

Early analysis of the debate was mixed, but the distillate of punditry certainly appeared to be that Obama, with his back up against the wall, had put things back on the right track, even if he didn’t move the his campaign’s train forward a great deal.

The spin by campaign operatives here began before the debate had even concluded, with both sides claiming victory.

“Whatever the president lost in his first debate, he more than made up for in this debate,” New York Sen. Charles Schumer (D) said. He added that if a similar number of people tuned in to this one as watched the first one, he thought people would see “a real bounce up for the president.”

Romney adviser Ed Gillespie insisted his man came away victorious. “The fact is, Gov. Romney won this debate because the American people heard from him clearly what he will do to turn this country around,” Gillespie said.

Aside from the Libya exchange, the president had other apparent victories in a debate filled with tension that ricocheted from the room here across television screens to millions of voters at home, but the hits weren’t as clean and Romney stood his own.

A testy exchange toward the beginning of the more than 90-minute event set the stage for the rough-and-tumble debate to come.

In response to a question from the audience about the increase in gas prices, Obama spoke about the increase of gas, oil and efficient car production. He said jobs were being created. “That’s the strategy you need, an all-of-the-above strategy, and that’s what we’re going to do in the next four years,” he said.

“But that’s not what you’ve done in the last four years,” Romney quickly retorted. “That’s the problem. In the last four years, you cut permits and licenses on federal land and federal waters in half.”

Obama cut in: “Not true, Gov. Romney.”

Romney asked the president how much he had cut.

“Governor, we have actually produced more oil,” the president said.

“No, no,” Romney pressed with some aggression. “How much did you cut licenses and permits on federal land and federal waters?”

Both began to talk over each other, standing just a few feet apart in a heated moment that appeared to reveal just how little either of the men like the other.

“I’m happy to answer the question,” Obama said.

Local newscasts late Tuesday were already playing that clip as one of the marquee exchanges of the debate.

Romney also had some very strong riffs, particularly discussing the Obama record. Perhaps his best played on the idea of whether voters are better off than they were four years ago.

An audience member asked what Obama had done over the past four years to earn his vote. The president went through a litany of promises he had kept and the things he had done to get the country moving again. Romney then took the stage.

“I think you know better. I think you know that these last four years haven’t been so good as the president just described and that you don’t feel like your confident that the next four years are going to be much better either,” he said. “I can tell you that if you were to elect President Obama, you know what you’re going to get. You’re going to get a repeat of the last four years. We just can’t afford four more years like the last four years.”