* Obama, both Bushes, Clinton and Carter to attend ceremony
* Political enemies with mutual respect
* The closest ties: Bill Clinton and George H.W. Bush
By Steve Holland
WASHINGTON, April 22 (Reuters) - Their rivalries have helped to define American politics for more than a quarter-century.
And sometimes the complex relationships among the only five people alive who know what it's like to be president of the United States have seemed to be straight out of a soap opera. They have called each other names and blamed one another for the nation's problems.
But when they have a rare meeting in Dallas on Thursday for the opening of former president George W. Bush's library and museum, there will be smiles for the cameras and friendly chatter by President Barack Obama and former presidents Bush, Bill Clinton, George H.W. Bush and Jimmy Carter.
It will be the first time they have all been together since January 2009, when they met in Washington a few days before George W. Bush left office and Obama was sworn in as president.
The library's dedication at Southern Methodist University will be something of a re-emergence for George W. Bush, who has preferred to stay out of the spotlight since leaving Washington after eight tumultuous years in office that followed six years as Texas governor.
"Fourteen years was enough for me," Bush told People Magazine last week. "But I do want to stay engaged in issues that matter to me."
Having served in a stressful job that always seems to age its occupants, the five men share a "common bond that supersedes any short-term policy or political differences," said Karen Hughes, a former top aide to George W. Bush.
That said, this isn't necessarily the friendliest of fraternities.
Obama, a 51-year-old Democrat, twice won election by denouncing Republican George W. Bush's handling of the presidency, from the struggling U.S. economy to wars in Iraq and Afghanistan.
Obama, the nation's first African-American president, also accused fellow Democrat Clinton of injecting race into the 2008 campaign, when Clinton was campaigning in South Carolina for his wife, Hillary, in the Democratic presidential primary.
Republican George W. Bush, 66, criticized Clinton's handling of the economy in defeating Clinton's vice president, Al Gore, in 2000. Clinton used the same economic argument in 1992 to deny a second term to Bush's father, George H.W. Bush.
It was George H.W. Bush, now 88, who had perhaps the most colorful putdown of Clinton and Gore in the 1992 presidential campaign when he said, "My dog Millie knows more about foreign policy than those two Bozos."
'YOU CANNOT GET MAD AT THE GUY'
It took a while for the edge to wear off from that campaign.
When then-President Clinton attended George H.W. Bush's presidential library opening in College Station, Texas in 1997, Clinton aides recall that Bush and his wife, Barbara, were particularly gracious.
On the other hand, their sons, George W. and Jeb, were frosty to the Clinton side, former Clinton press secretary Mike McCurry said.
Five years after Clinton had denied the elder Bush a second term in the White House by casting the Republican as out of touch on the economy, "things were still a little bit raw," McCurry said.
Now, Clinton and George H.W. Bush probably have the closest friendship within the group of presidents. They have worked together to raise money to help those stricken by a 2005 tsunami in the Indian Ocean that killed more than 230,000 people in more than a dozen countries.
"You cannot get mad at the guy," George H.W. Bush wrote of Clinton after traveling with him in 2005. "I admit to wondering why he can't stay on time, but when I see him interacting with folks my wonder turns to understanding, with a dollop of angst thrown in."
The younger Bush now gets along with Clinton, too. They worked together on Haiti earthquake relief in 2010.
"I like him, and I love his father," Clinton said during an appearance with George W. Bush in Salt Lake City last August.
The relationship between the last two presidents, Obama and George W. Bush, remains something of a work in progress, aides said.
The two rarely speak, although Obama was gracious to his predecessor last year when Bush visited the White House for the formal unveiling of his presidential portrait.
"We may have our differences politically, but the presidency transcends those differences," Obama said at the ceremony. "We all love this country. We all want America to succeed."
A senior administration official said that Obama, now in his second term, has not changed his views about what he saw as "poor policy decisions" by Bush on the Iraq war and the U.S. economy. But, the official said, Obama has an appreciation "for the enormity of the decisions that a president has to make and the burden that a president has to bear, especially when Americans lose their lives."
The Bush side, which was not happy at how Obama bashed him during the 2008 campaign, nevertheless is pleased that Obama is joining the group in Dallas.
If there is a wild card in the group, it is Carter.
Aides to Obama and those close to both Bushes and Clinton say the four of them are all a bit baffled and bemused by Carter, 88, who was president from 1977 to 1981, before the late Ronald Reagan's conservative revolution stormed Washington.
Carter has consistently criticized all of his successors, even when they have been fellow Democrats.
"The odd man out is Carter," said Ron Kaufman, a former adviser to the elder Bush.
So what do the presidents talk about when they get together? If the past is any guide, they will keep it light.
Dana Perino, who was a press secretary to George W. Bush, said after the presidents were together in 2009 she asked her boss what they had discussed.
Bush's response: "We mostly talked about our families." (Editing by David Lindsey and Vicki Allen)