July 29 (Bloomberg) — Washington being Washington, the hottest relationship in town doesn’t revolve around sex or even the next presidential election: it’s the political courtship of old antagonists, Barack Obama and John McCain.

Political relationships, especially those involving the president, are the sustenance of the American capital. Sometimes they are poisonous: President Lyndon Johnson and Robert F. Kennedy, as captured in the latest volume of Robert Caro’s biography of LBJ. At other times, they are lopsided, as when Bill Clinton dominated Newt Gingrich under the guise of working together. Every now and then, there are adversarial/symbiotic relationships that, on balance, get things done: Ronald Reagan and Tip O’Neill in the 1980s, for example.

The association between Obama and McCain is different. But it may be Washington’s most important since Reagan and O’Neill.

McCain, 76, whose political resiliency is rivaled only by such luminaries as Bill Clinton and Richard Nixon, is the most pivotal figure in the Senate today. He often is more central than the party leaders, Mitch McConnell, the Kentucky Republican, or Harry Reid, a Nevada Democrat, or the self-styled new power broker, the New York Democrat Chuck Schumer.

When McCain is with the president — on immigration and in brokering the recent deal to secure Senate approval of stalled Obama nominees — they usually can trump the political right. When he’s against him — sabotaging Obama’s plan last year to nominate Susan Rice as secretary of state — the White House rarely prevails.

Their previous strains predated 2008, when they vied for the presidency. Obama saw his Republican rival as an embittered, compromised maverick who treated him as an undeserving upstart. That was close to the mark. After he lost that election, McCain saw Obama as naïve, aloof and surrounded by too many sycophants.

In 2011, there was a move to détente after Arizona representative Gabrielle Giffords was shot. That, however, was “a false start,” McCain recalled in an interview last week.

This time, political convenience broke the ice. A re-elected president soon realized that without the support of a small core of Senate Republicans, any agenda was doomed. McCain, who moved right to fend off a Tea Party primary challenge in 2010, was itching to reclaim his maverick persona and wage a two-pronged battle: against the isolationists and political right of his own party and against the national-security left wing in the Democratic camp.

Since the January inauguration, Obama and McCain have met a dozen times. In half of those occasions, they were either alone or with only a few other principals. Although the discussions were usually about immigration policy, they invariably ranged more broadly.

There are huge tests ahead, especially the budget/debt ceiling/sequestration battles this autumn. McCain, the defense hawk, despises the across-the-board cuts to defense and discretionary domestic spending required under sequestration and wants to help forge a compromise replacement involving more taxes and cuts in entitlements.

The odds are against that happening; most House Republicans are more eager for an economy-threatening standoff than for an accord. The only slim hope is a deal, led by the White House and a small group of McCainites.

McCain also wants to help Obama fulfill his promise to close the detainee camp for terrorism suspects at Guantánamo Bay, Cuba. He says political conditions are much different from four years ago, when there was a similar effort.

“The difference between 2009 and 2013 is the administration now has a plan,” he says.

Last month, the five-term senator traveled to Guantánamo with Senate Intelligence Committee chairman Dianne Feinstein and the White House chief of staff, Denis McDonough.

McDonough, who McCain knew as a mid-level aide to former Democratic Senate Leader Tom Daschle, is a glue that binds the Republicans and the administration. He and McCain talk as often as five times a day. In addition, the Republican senator has a great fondness for Vice President Joe Biden, a good working relationship with Secretary of State John Kerry and is a fan of United Nations Ambassador-designate Samantha Power.

He’s also an unrelenting critic of much of Obama’s foreign policy. He sees the president as indecisive or soft on Syria, Egypt and Afghanistan. He has a long running feud with General Martin Dempsey, the chairman of the Joint Chiefs of Staff.

He has no regrets about torching Rice’s appointment to be secretary of state, and continues to suggest that she dissembled on the terrorist attack last year against a U.S. mission in Benghazi, Libya. But when she was tapped to head Obama’s National Security Council, McCain wrote a Twitter post saying that he’d make “every effort” to work with her. She contacted him, they had a cordial meeting and he asked her to discuss Syria with one of his confidants, retired General Jack Keane. Rice met with Keane: “I can’t ask for anything more,” McCain says.

He’s as eager to take on the Rand Paul isolationists within his party. In an April speech at the Center for a New American Security, he focused criticism on his fellow Republicans. He has disdain for much of the movement right and little regard for Senate Majority Leader McConnell.

In the interview in his Capitol hideaway office last week, the never combat-shy McCain seemed to revel in his reclaimed persona and his multifront battles. He suggests, however, that his style is changing.

“The biggest mistake I used to make was getting personal,” he says, declaring that that his role model now is the late Democratic senator Ted Kennedy, who was a friend.

One prominent Democrat, who knows both Obama and McCain well, is certain the senator genuinely wishes to work with the president. Noting that for all the assistance of McDonough or even Biden, at this level, interactions ultimately depend on principal-to-principal connections, and this politician worries that Obama doesn’t do relationships well.

Asked about that, McCain paused, then acknowledged that the president isn’t a “schmoozer” like his predecessors Clinton and Reagan.

“He’s willing to compromise but sometimes not sure he knows exactly how to do it,” McCain says. “We can help.”

(Albert R. Hunt is a Bloomberg View columnist. The opinions expressed are his own.)

Photo: Gage Skidmore via Flickr.com