WASHINGTON — On the seal of the United States, the eagle turns its head toward its right talon, which holds an olive branch, and away from the talon holding 13 arrows. It is meant to suggest a preference for peace. The eagle that hovered between the two candidates in the second debate had the same design, but for one difference: The eagle's head was turned toward the arrows. It was a fitting symbol for the pointed and sniping contest between President Barack Obama and Gov. Mitt Romney Tuesday night.

 It was a night of barbs, interruptions and charges and counter-charges. "Very little of what the president said is true," said Romney. "It's not true, governor," said Obama. "Not true. It's not true." During one exchange, Romney said, "You'll get your chance. I'm still speaking," as the audience in the arena seemed to gasp. At another point, the two men got so close and huffy, I thought moderator Candy Crowley might just ask them to take it outside.

It's a pretty good bet that undecided voters missed the important matter the guys were squabbling over - was it about a barbecue grill or the Lions and Bears football game? The debate was the only town hall-style contest in which voters posed questions. Just as the candidates seem to be speaking past the electorate, it often felt as if they were speaking past the questioners. Both candidates avoided answering questions about gas prices. Romney continued to show an aversion to specifics in explaining his tax plan, and the president didn't do a great job of answering a voter who wanted to know what Obama was going to do to earn his vote.

If you had to score a winner (and I am obligated by the memorandum of understanding to do so), you'd give the night to the president - barely. He was a forceful advocate for his policies, kept Romney on defense, and gave Democrats plenty to cheer. He probably stopped his slide in the polls. He looked like a person who would fight for the middle class.

In the first debate, he had looked like a person who wanted to skip middle class and get out to recess. In Round 2, Obama's closing remarks about fighting for the 47 percent of the popular vote that Romney disparaged in that secret video were among his best of any debate he's done on the national stage. When you finish strong, you are usually in a good place.

But Mitt Romney showed that his first debate performance was no fluke. He gave as good as he got, and he consistently offered a confident alternative to the weak Obama recovery. "I can tell you that if you were to elect President Obama, you know what you're going to get," he said. "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."

Both candidates had bad moments: President Obama tried to reinvent his initial reaction to Libya in front of millions of people, and Romney tried to tell a story about women in his cabinet that was certainly off-key and perhaps not even true.

In a CBS poll of undecided voters after the debate, the group split nearly in thirds, with Obama winning slightly with 37 percent over Romney's 30 percent. Thirty-three percent thought the contest was a tie. Voters in the CBS poll by almost 3-to-1 thought Romney could better handle the economy, but the president won by a large margin on the question of which candidate cared about the middle class. A poll of "Walmart Moms" gave the edge to the president, but not by much. The CNN poll gave the president a victory by 7 percentage points. A race that was tied - or at least very close - may now be even more tied than ever. (Tied with a double knot?) It is as if the ups and downs of the last several months never happened.

In a conversation with one of Obama's top lieutenants in Ohio last week, I asked what he could say in the second debate that he didn't say in the first. "Outsourcing, offshoring and autos," was the response, referring to Bain Capital's investment in companies that outsourced American jobs, Romney's use of offshore accounts, and his refusal to back the auto bailout. The president hit all those themes in the first 15 minutes. He also tied Romney to the very unpopular Republican Congress. He had the Midwest in mind when he mentioned his plan to create more manufacturing jobs, and the debate over the car bailout put Romney on the defensive.

Each candidate accused the other of not telling the truth, and both of them offered us examples of what that looks like. Romney told a false story about the past, and Obama told a false story about the present. During a politically charged exchange about fair pay for women, Romney recounted the early days of his Massachusetts administration. "I had the chance to pull together a cabinet, and all the applicants seemed to be men," Romney said. And I - and I went to my staff, and I said, 'How come all the people for these jobs are - are all men?' They said, 'Well, these are the people that have the qualifications.' " In recounting how he went on a search for qualified female candidates, Romney remembered, "I went to a number of women's groups and said, 'Can you help us find folks,' and they brought us whole binders full of women."

On its face, the answer is odd. Romney sounds a little out of touch with the inequities of the workplace. The notion that women had to be scared up from the bushes seems from another era. But the main point for a candidate whose veracity is a key talking point for his opponent is that the story may not be true. The Boston Phoenix has fact-checked the story and claims that the binder of women was presented to Romney when he came into office. He didn't appear to ask for it, as he claimed in Tuesday night's debate. Perhaps he asked for another binder. In the end, however, as governor, Romney did hire a lot of women.

Romney was asked directly about how he was going to pay for his tax plan, and his answer was no better than it has been for months. He named no loopholes he would close but offered the faux-specificity of a $25,000 cap on deductions. This is a very small piece of specificity in explaining how he would pay for a $5 trillion tax cut.

 "Governor Romney was a very successful investor," said the president. "If somebody came to you, governor, with a plan that said, here, I want to spend $7 or $8 trillion, and then we're going to pay for it, but we can't tell you until maybe after the election how we're going to do it, you wouldn't take such a sketchy deal and neither should you, the American people, because the math doesn't add up."

The president's tall tales came during a debate over Libya. The administration's story is changing almost daily about what happened, who knew what when, and who is going to take responsibility for it. The topic presents political peril for the president. He effectively took command, saying that all responsibility rested with him and that he would get to the bottom of who killed the four Americans, including the ambassador. He criticized Romney for using the issue to score political points and took umbrage at the suggestion that anyone in his administration would act politically.

Then, he proceeded to act politically. That is, if you define acting politically as suggesting something that isn't true is true for the purpose of saving your job.

Obama said, "The day after the attack, governor, I stood in the Rose Garden and I told the American people . . . that this was an act of terror." Obama was trying to suggest that he had declared this a terrorist attack long before his administration actually did. For days and days afterward, administration officials would not claim it was a terrorist attack.

U.N. Ambassador Susan Rice famously refused to call it such on "Face the Nation." The president was trying to reset the timeline. If you look at the president's statement in the Rose Garden, he does use that phrase, but it's a throwaway cliché. Indeed, it arguably wasn't about the attack at all, just a bromide about more general acts of terror.

In any event, the president buried the lead in the tenth paragraph of his remarks. That's why none of the papers at the time reported that he had characterized any part of the attack as having to do with terrorism. When Romney called him on it, the president wouldn't answer. "Please proceed, governor," he said, as if he were the moderator and not the fellow who was being called out. It was the verbal equivalent of putting your hands over your eyes and pretending no one will see you.

In the end, this picture of the debate says it all - the two candidates pointing at each other in mid-sentence. The next debate is less than a week away. The topic will be foreign policy and national security. The eagle will hang between the two men there, too. But for that final debate, questions of war and peace will actually be central to the discussion, not just a fitting description of it.

              

Dickerson is Slate's chief political correspondent. On Twitter: @jdickerson