Last summer, as Bill and Hillary Rodham Clinton celebrated the former president’s 65th birthday with a party at their rented Hamptons home, talk among their guests turned to President Barack Obama’s travails over the debt crisis and doubts about his re-election.

"I’m really trying to help him," the white-haired former president said, shaking his head, "but he seems to have lost his narrative."

Starting Tuesday, that help will take its most public form yet, in the form of a new book by Clinton, titled "Back to Work," that makes a case for confidence in government and Obama’s jobs bill. The book marks a new and somewhat warmer stage in the two men’s rivalry and relationship, one that could benefit Obama. The awkward twist: The former president has been so frustrated at what he sees as the current one’s failure to explain his economic policies that he has literally decided to write his own version of the story.

"He both sells it and wants them to sell it more," said John Podesta, a former Clinton chief of staff who also advises the current administration.

The differences between the two leaders go to the very bedrock of personality and temperament. Obama has discipline that Clinton can barely dream of; Clinton excels precisely where Obama struggles (at economic storytelling and attracting white working-class voters). Their divide began long before Obama’s 2008 primary battle with Hillary Clinton, now the secretary of state. In 1996, the new Illinois state senator criticized Bill Clinton’s move to the center, called his campaign "disturbing" and said his convention was "for sale"; Clinton returned the favor by endorsing Rep. Bobby Rush, D-Ill., whom Obama had challenged.

There has been no recent heart-to-heart talk for the two men, say their aides, no confessional moment about the ill will generated during the 2008 campaign. Instead, they have improved their bond through a series of mutual allies — Hillary Clinton, but also Vice President Joe Biden, their chief mediator — brief but positive meetings, and shared beliefs.

The Obamas have never invited the Clintons for dinner at the White House, but there have been other encounters like a recent golf game and an impromptu joint briefing room appearance in 2010 and a private 2011 meeting in the Roosevelt Room that generated an energy-efficiency initiative. They are fundamentally united on policy issues, which helps.

"The policy has reinforced the personal," said Gene Sperling, a top economic adviser to both administrations, who briefed the eager former president on the jobs bill when Obama announced it.

At home in Chappaqua, N.Y., Clinton spends more than an hour a day "studying the economy," as an aide put it, meaning clipping newspaper articles and reading obscure government reports.

"I think he thinks to himself, ‘God, this is what I would do"’ in response to the grim circumstances, a former Clinton economic official said.

"Back to Work" is a classic Clinton talkathon, zooming from job-creation ideas to a complaint about something he saw on Bill Maher’s show. He expresses sympathy for Obama, but his private critique comes through as well: incredulity that the president and Democratic leaders did not raise the debt ceiling during the 2010 lame-duck session; bafflement that many beneficiaries of Obama’s policies "didn’t even know about" his actions; and frustration about the lack of a powerful, unified Democratic message in the midterm elections.

He said that he and Biden failed to persuade the Democratic National Committee to send out talking points back then. ("We appreciate his insights and his advocacy," David Axelrod, Obama’s chief strategist, said in an interview Friday.) Clinton said he decided to write the book after he campaigned in the midterm elections and met voters who thanked him for explaining the administration’s policies in a way they had never understood before. He appeared at 130 campaign events, he notes, without adding that Obama was criticized by fellow Democrats for not doing enough.

Clinton has never gone light on self-celebration: His birthday party last month was a three-day star-studded Hollywood celebration (the Hamptons one was just a prelude). "Back to Work" is his latest attempt at legacy building. Last week, at a rosy retrospective of Clinton-era economic policy mounted by his foundation, the former president and his aides, including some who serve in the Obama administration, tried hard, several admitted later, to salute the former president’s record of balanced budgets and job growth without damning the current one’s.

Almost every speaker stipulated that Obama’s challenges — the worst economic slump since the Great Depression, and Republicans even less willing to yield than they were in the 1990s — were far greater than Clinton’s. And yet the slickly produced program, including videos and handouts with titles like "Looking Back at President Clinton’s Achievements for What We Can Learn from Them Today," seemed like an implicit rebuke to Obama: The 42nd president bested Republicans in a budget showdown, far outperformed his own economic projections, slammed the unemployment rate down to 3.9 percent and stuck to an economic message.

Still, Clinton finds the beleaguered Obama of 2011 easier to sympathize with than the rising star of 2008, aides and friends say. In September, the ex-president defended Obama at a private memorial service for Howard Paster, his former congressional liaison.

"I get tickled when I hear people talk about how President Obama should do this, that or the other thing," he said. "We had a mostly Democratic Congress, and we were committed to reversing trickle-down economics and we could only pass the plan by one vote in the House and Senate."

Obama and his aides, meanwhile, have little choice but to put up with the former president’s reminders of his achievements, because of the hard political reality: Clinton can potentially help them a lot in 2012.

"Clinton does something that’s almost unique in American politics, which is talk to the middle," said Bill Carrick, a Democratic political consultant who worked for him decades ago.

For the record, Obama won the recent golf competition. The photographs from the excursion tell the same story the friends do about the outing: The older president is open-mouthed and gesticulating. The younger one looks like he would rather just play golf but is patiently listening anyway.