In their 1976 presidential debate, Jimmy Carter criticized President Gerald R. Ford for his choice of running mate, for his energy and environmental policies and for what he described as “the failures of this administration,” adding, “I would do the opposite in every respect.” Years later, Carter and Ford would become close friends, bound together, as Carter put it at his predecessor’s funeral, “for the rest of our days.”

In the 1992 presidential campaign, Bill Clinton pilloried George H.W. Bush as the symbol of the out-of-touch plutocrat who didn’t understand the stresses affecting the average American. A decade later, the two were so inseparable that Barbara Bush joked that Clinton was “like a son” to her.

In the year 2000, George W. Bush ran for the White House vowing to restore the dignity of the presidency, a swipe at Clinton. Now we know that Bush called Clinton from the White House from time to time, and earlier this month, the two former presidents appeared in Barack Obama’s White House, teaming up to help the people of Haiti after a devastating earthquake threw the Caribbean nation into chaos.

That recent scene in the capital was remarkable — and at the same time unremarkable. There were the two most recent presidents, opposites in almost every respect — one liberal and the other conservative, one extroverted and the other introverted, one who raised taxes and the other who lowered them. And yet what threw them together was stronger than what had separated them, which is why so many former rivals find themselves to be allies once they are former presidents.

“You learn a lot about a man when you run against him for president,” Carter said at Ford’s funeral, “and when you stand in his shoes and assume the responsibilities that he has borne so well, and perhaps even more after you both lay down the burdens of high office and work together in a nonpartisan spirit of patriotism and service.”

No one who knew both men was surprised to see Carter delivering a eulogy to his predecessor. Indeed, it was not uncommon, once he had left office, for Carter to make a conversational point and then to add that “President Ford,” as he always called him in public, surely would have some important further thoughts on the matter.

All presidents are different, to be sure, but they share the same burdens, pressures and lonely moments of decision. And once out of office they often share the same podiums, in part because they are members of a very exclusive alumni association.

“There is no training ground for being president, nor is there any camaraderie once you leave the White House, except for the very small group of people who have been in the White House,” says Shirley Anne Warshaw, a presidential historian at Gettysburg College. “This is one of the smallest peer groups in the world.”

Small — but powerful. Which is why there was more than symbolism, more even than a statement about continuity and comity in American civil life, in the sight of three presidents — Obama, Bush and Clinton — in contemplative conversation outside the White House. This was a bipartisan American expression of concern — and resolve.

Indeed, photographs showed their heads bowed at about the same introspective angle as John F. Kennedy and Dwight D. Eisenhower in the Pulitzer-winning photograph taken by Paul Vathis of The Associated Press shortly after the Bay of Pigs fiasco in 1961. “They looked so lonely, the young president and the older war hero, and the stripped trees in the background, two men burdened with major troubles,” Vathis said, according to an AP account.

Suddenly it didn’t matter that Kennedy had won the White House only months earlier on a pledge to “get America moving again,” an implicit criticism of the Eisenhower administration. Nor did it matter this month that Bush had been a relentless critic of Clinton, nor that Obama had spent nearly a year blaming the economic crisis on Bush. “Now’s not the time to focus on politics,” Bush said on the CBS program “Face the Nation,” adding, “It’s time to focus on helping people.”

He and Clinton offered a noteworthy coda, in the form of an op-ed piece in The New York Times, in which they said that “throughout both our careers in public service” they had “witnessed firsthand the amazing generosity of the American people in the face of calamity,” being careful to mention calamities during both of their presidencies. But what was stunning was the political statement they appended to their appeal for funds:

“We should never forget the damage done and the lives lost, but we have a chance to do things better than we once did; be a better neighbor than we once were; and help the Haitian people realize their dream for a stronger, more secure nation.”

That was the venue in which Bush said that his mother considered Clinton like a son. “So, brother,” Bush said, “it’s good to see you.” In many ways, it’s good to see them together.