At their prime, John Adams and Thomas Jefferson were vicious rivals with sharply divergent visions of the kind of nation America should be. Their exchange of letters and thoughts in their dotage is one of the great collections of correspondence in all of history. Gerald Ford and Jimmy Carter so disliked each other that the tension at their presidential debate 30 years ago was palpable. Once out of office, the two grew so close that it became hard to know what the fuss in 1976 was all about.

Now George H.W. Bush and Bill Clinton, rivals in 1992 and, in a proxy war between one man’s son and another’s vice president, in 2000, have drawn so close that President George W. Bush last week referred to his predecessor as “my new brother.”

It wasn’t always that way, of course. Mr. Clinton used to refer to the president’s father as “old Bush” and portrayed the 41st president as an out-of-touch, out-of-his-element symbol of privilege. The tension between the two families only increased as the younger President Bush sought to regain the office that Mr. Clinton forced his father to relinquish. Both Bushes spoke vaguely of the need to preserve the dignity of the office of president, and no one mistook their meaning, which was that the Bushes moved to the strains of classical music while the Clintons lived their lives to the twang of a country-western ballad.

Once in the office that he repeatedly suggested Mr. Clinton had callously defamed, the new president almost never referred to Mr. Clinton by name. In the first several months of the Bush 43 era, it seemed as if the president were more concerned about not being Bill Clinton than he was about being George W. Bush.

In truth, the Bush presidency defined itself by not being the Clinton presidency. Every meeting started on time. (Mr. Clinton’s had started on time only if the margin of error was 90 minutes.) The president went to bed early. (Mr. Clinton stayed up late talking politics and talking to politicians.) The White House was leak proof. (Mr. Clinton’s was a colander.) The first lady was quiet, unopinionated, secure in her Texas roots. (Mrs. Clinton tried to take control of one-seventh of the American economy and, that having failed, ran for the Senate from a state she had hardly visited before.) The president was monogamous. (Mr. Clinton believed in marriage but was not a fanatic about it.)

Then there was the substance. In the Clinton years, taxes went up. In the Bush years, taxes went down. In the Clinton years, the United States made a point to consult with allies and with international organizations. In the Bush years, the allies and the United Nations found out about White House intentions by tuning into CNN. In the Clinton years, the end of the era of big government was proclaimed. In the Bush years, the government, bloated by the Department of Homeland Security and then the belated response to Hurricane Katrina, grew like mad. (Nobody said that either man stayed on the script.)

Indeed, it is difficult to think of contiguous presidencies that have been so different, at least since Warren Harding, who was no intellectual, followed Woodrow Wilson. As president of Princeton, Wilson behaved as if he were president of the United States; but when he became president of the United States, he behaved as if he were the president of Princeton. Franklin Roosevelt was substantially different from Herbert Hoover, of course, but it is hard to say that the Reconstruction Finance Corp. that Hoover fashioned to fight the Great Depression did not foreshadow the New Deal. (Can’t wait to get the angry e-mails on this one.)

So why has the Bush family drawn so close to the Clinton family – and vice versa? It is not only because Mr. Clinton both followed and preceded a President Bush and because it is now possible that President Bush could both follow and precede a President Clinton. (Think about it.) Nor is it because Mr. Clinton never had a father, or rather never knew his father, who died before he was born, and never had a full brother. (Imagine what Fawn Brodie, the historian who psychoanalyzed Thomas Jefferson, would have to say about William Jefferson Clinton.)

Mr. Clinton and the elder Mr. Bush were thrown together when the younger President Bush asked them to work together to assist the victims of the Asian tsunami. Once they became the 21st-century version of Butch Cassidy and the Sundance Kid, they got together again when the twin hurricanes hit the American Gulf Coast last year. They realized that they shared a bond and a burden that fewer than four dozen Americans have shared in more than two centuries.

“I can understand why ex-presidents are able to put aside old differences,” the president said in a CBS interview last week. He also said: “One of these days, I’ll be a member of the ex-presidents’ club. … I’ll be looking for something to do.”

George W. Bush’s passage from president to ex-president will test the theory that presidents (with the exception of Harry Truman and Dwight Eisenhower, wary of each other from the start, and Theodore Roosevelt and William Howard Taft, whose relationship and succession drama was worthy of Greek literature) eventually regard each other as members of one big happy family. Will three be a crowd? Will the bookish Mr. Clinton (a Rhodes Scholar) and the thoughtful President Bush 41 (a member of Phi Beta Kappa at Yale) welcome President Bush 43 (a member of Delta Kappa Epsilon at Yale)? And, if it comes to this, how will Bush 41, Bush 43 and Clinton 42 respond if they get the call to duty from … Clinton 44? The possibilities in this family drama are endless.