Democratic presidential hopefuls Sen. Barack Obama, D-Ill., and Sen. Hillary Rodham Clinton, D-N.Y., joke Thursday at the end of a Democratic presidential debate in Los Angeles.

Los Angeles ? Democrats Barack Obama and Hillary Rodham Clinton sparred, for the most part cordially, over immigration, health care and the war in Iraq in their first one-on-one debate on Thursday as they faced high-stakes Super Tuesday contests that could go a long way toward determining the party’s presidential nominee.

Clinton emphasized that the nation needed a president ready to go to work on “Day One.” Obama responded: “Part of the argument that I’m making in this campaign is that it is important to be right on Day One.”

Five days before Super Tuesday, the two alternated between civility and pointed swipes, underscoring the importance of the upcoming contests.

Clinton defended the increasingly high-profile role of her husband, former President Clinton, in her campaign and his recent sharp criticism of Obama. “At the end of the day, it’s my name that is on the ballot.”

Both were asked about the possibility of a “dream ticket” of Clinton-Obama – or Obama-Clinton.

“Obviously there’s a big difference between those two,” Obama said. “I respect Senator Clinton; I think her service to this country is extraordinary.” But he said, “We’ve got a lot more road to travel” before such a decision.

Clinton agreed it was too early to discuss running mates.

Both predicted that one of them would be the next president in a history-making inaugural. Obama would be the first black president, Clinton the first female president.

Clinton said the Republicans are “more of the same” and, gesturing toward Obama, she said, “We will change our country.”

Making amends for his apparent snub of her at Monday’s State of the Union Address, Obama assisted Clinton by pulling back her chair as the debate – televised on CNN – began and ended. They then embraced.

But it wasn’t all sweetness and light.

One of their most pointed exchanges came on the question of whether illegal immigrants should be able to obtain driver’s licenses. Obama supports doing so; Clinton initially supported it and now opposes it.

“Senator Clinton gave a number of different answers over the course of six weeks on this,” Obama said, turning to Clinton. “Initially, you said you were for it, then you said you were against it.” He said he was raising her wavering to underscore that it is “a difficult political issue.”

Clinton called the controversy “a diversion” from efforts to come up with comprehensive immigration reform. “I co-sponsored immigration reform in 2004 before Barack came to the Senate,” she said.

Obama argued for his candidacy, saying, “I respect Senator Clinton’s record. I think it’s a terrific record. But I also believe that the skills that I have are the ones that are needed right now to move the country forward, otherwise I wouldn’t be running for president.”

They also clashed on Iraq.

Clinton suggested only she had “the necessary credentials and gravitas” to lead the country in withdrawing from Iraq without endangering U.S. forces or further destabilizing the region. She said it was crucial to bring Syria and Iran to the diplomatic table.

Obama shot back: “Senator Clinton mentioned the issue of gravitas and judgment. I think it is much easier for us to have the argument when we have a nominee who says ‘I always thought this was a bad idea. This was a bad strategy.’ It was not just a problem of execution.”

Clinton voted in October 2002 to authorize President Bush to use force in Iraq, while Obama opposed such authority in a speech he gave in 2002 as a member of the Illinois state Senate.

The two also reached out quickly to backers of former rival John Edwards, who bowed out of the race Wednesday without endorsing either one. Both praised his efforts in their opening statements.

Asked whether it was good for the country to have another Clinton in the White House, further extending Bush and Clinton family control over government, Clinton drew applause in the Kodak Theatre – home of the Academy Awards – when she replied, “It did take a Clinton to clean after the first Bush and I think it might take another one to clean up after the second Bush.”