Bill Clinton and Barack Obama have not always seen eye-to-eye. Here are seven defining moments in their relationship. (The Washington Post)

The pair of presidents settled into their plush armchairs Tuesday, crossed their legs and tried to demystify the new health-care law, which, as President Obama explained, has become “a little political.”

On this afternoon, on the glistening stage of his annual charitable gathering, former president Bill Clinton asked the questions. And Obama, as is often the case, wasn’t short with his answers.

The two men who stand as bookends for the modern Democratic Party made a united sales pitch to millions of uninsured Americans to enroll when new insurance marketplaces open Oct. 1.

“I don’t have pride of authorship for this thing,” Obama said of the law that could determine his legacy. “I just want the thing to work.”

Clinton added: “I think this is a big step forward for America. But first, we’ve got to get everybody to sign up.”

Obama has enlisted Clinton again as his “secretary of explaining stuff,” a nickname he earned after his well-received speech at the 2012 Democratic National Convention. The aim is for Clinton to help sell the health-care law to skeptics across the country while combating Republican attempts to undermine it.

Their joint appearance Tuesday afternoon at the Clinton Global Initiative in New York came as the White House launches an aggressive public relations campaign to encourage people to learn more about their coverage options under the law and enroll in the insurance marketplaces.

The two presidents — whose relationship has warmed since its low point during the bruising 2008 Democratic primary battle between Obama and Clinton’s wife — were introduced by Hillary Rodham Clinton. The former secretary of state playfully talked about the similarities between her husband and her former boss, noting that both are left-handed, enjoy playing golf (“a game that does not often reciprocate the love they put into it”) and are “master politicians” who each have lost just one election.

Plus, she added, “They each married far above themselves.”

Earlier in the afternoon, Hillary Clinton spoke on a panel on “advancing progress” for women and girls and said she believes that electing the first female U.S. president “would be a very strong statement.”

“Someday, I hope it happens,” she said, adding that her remark was not about her own political ambitions.

In front of a crowd of boldface names, including former secretary of state Madeleine K. Albright and Olympic champion figure skater Scott Hamilton, Bill Clinton and Obama talked extemporaneously about health care for nearly an hour.

Clinton began by asking Obama to explain why, in the midst of a deep and painful recession during his first term as president, he pushed so hard for a complicated health-care overhaul.

“I think it’s important to remember that health care is the economy — a massive part of our economy — and so the idea that somehow we can separate out the two is a fallacy,” Obama responded. He went on to argue that the overhaul has economic benefits for businesses and the government’s deficit.

“That’s about as good an overview as you’re going to hear,” Clinton said.

Obama and Clinton acknowledged the political difficulties Democrats have faced in promoting the merits of the law amid strong GOP opposition. Normally, Obama said, the implementation phase is “pretty straightforward.”

“But let’s face it,” he added, “it’s been a little political, this whole Obamacare thing.”

The president’s remarks in New York came as Sen. Ted Cruz (R-Tex.) mounted a filibuster-style demonstration in Washington over his opposition to Obamacare. Cruz and other Republicans want to strip funding from the overhaul in exchange for keeping the government open after an Oct. 1 shutdown deadline.

“One of the things you and I both know is that when [it comes] to health care, there’s no more personal and intimate decision,” Obama said to Clinton. “. . . The devil you know is always better than the devil you don’t know, and that’s what ‘Harry and Louise’ was all about back in the 1990s. It was scaring people with the prospect of change.”

Clinton smiled at Obama’s reference to a pivotal television ad campaign that helped derail his own attempt to pass a health-care law in the first years of his presidency.

He prodded Obama to highlight some successes with the new insurance marketplaces in the states — particularly Clinton’s own.

“Tell ’em about Arkansas, because we’re doing well down there,” said Clinton, a former Arkansas governor.

“A little hometown bias,” the president responded. “Nothing wrong with that.”

Obama said three politically conservative states where “I just got beat” — Arkansas, Idaho and Kentucky — have set up exchanges and are seeing some success. He said that when people consider their coverage options under the new law, they will discover that it is “a good deal.”

“Do we want to continue to live in a society where we’ve got the most inefficient health-care system on Earth, leaving millions of people exposed to the possibilities that they could lose everything because they get sick?” Obama asked. “I think the answer is no.”