(CNN) A parting lesson in American civility from Sen. John McCain lies in the roster of leaders he personally selected to pay tribute at his memorial service Saturday at the National Cathedral.

It was a day in early April when Barack Obama received an unexpected call from McCain, who was battling brain cancer and said he had a blunt question to ask: Would you deliver one of the eulogies at my funeral?

Obama, who is responsible for extinguishing McCain's second bid for the White House a decade ago, immediately answered that he would. He was taken aback by the request, aides say, as was George W. Bush, another former rival, who received a similar call from McCain this spring.

When the 43rd and 44th US presidents stand on the high altar of the soaring cathedral on Saturday, after the "Battle Hymn of the Republic" anthem is sung, they will not only be celebrating the life of John Sidney McCain III. It will be McCain, too, having a not-so-subtle last word, aimed at another president he made clear he did not want to attend: Donald J. Trump.

While neither of the two former presidents were especially close to McCain in life, he and Bush were fellow Republicans, forged together for better or worse, through policy and party loyalty. After a deeply personal and vitriolic primary fight in 2000, McCain went on to endorse Bush and occasionally campaigned with him four years later.

But McCain's decision to invite Obama to speak at his funeral stands out as far more extraordinary, given their brutal and bitter rivalry during the 2008 presidential race.

I had a daily ringside seat to their feud, covering the campaign for The New York Times, chronicling their fights over the Iraq war and, later, the economy. On those subjects, and many more, McCain viewed Obama as naïve and unprepared for the presidency. To be clear, those critiques lingered long after Obama won, particularly on matters of national security.

So, I've been wondering whether McCain and Obama had somehow developed an intimate relationship after Obama left office, if they had been having quiet conversations over the last year or two that haven't been publicly discussed as McCain neared the end of his journey.

It turns out, after talking to several friends of both men this week, their relationship isn't intimate at all, but rather one rooted in mutual respect and a shared sense of alarm at today's caustic political climate. Their telephone call on that April day was first arranged by advisers, not McCain simply dialing up Obama as he would do with his legion of friends, a sign they were hardly tight.

In fact, the two have spoken by phone only a couple of times since Obama left the White House, aides to both men say, most notably last summer when Obama reached out after McCain cast the deciding vote to salvage the Affordable Care Act. He thanked him. The call was brief.

Obama has not been among the long parade of visitors who came to see McCain on his Arizona ranch as he fought brain cancer. George and Laura Bush dropped by not long ago, as did former Vice President Joe Biden, a close and longtime friend of McCain's in the Senate, who will deliver a eulogy at a memorial service on Thursday in Arizona.

But McCain's decision to ask Obama and Bush to eulogize him is part of a carefully choreographed -- and, yes, even strategic -- message for America and the world in the wake of his death. It's also perhaps, one last opportunity for McCain to try and tamp down a fervor that first awoke in the Republican Party during his 2008 race and has swelled ever since.

'A lesson in civility'

Steve Duprey, a longtime friend of McCain's and a senior adviser in his 2008 campaign, said the senator respected Obama, even if the two were never particularly close and wounds from their race were raw for years.

"I think it is John McCain imparting a lesson in civility by asking the two men who defeated him to speak, as an example to America that differences in political views and contests shouldn't be so important that we lose our common bonds and the civility that is, or used to be, a hallmark of American democracy," Duprey said.

David Axelrod, a senior adviser to Obama in his campaigns and in the White House, said the clear message McCain is sending is "about our shared heritage, our shared trust of this democracy that transcends party and transcends tribe."

"It really does animate his message of national unity," Axelrod said. "There is a kind of poetry to it that he wanted his two erstwhile opponents to eulogize them."

Even unspoken, the lesson also shines a light on McCain's outward disdain for Trump and his presidency. And McCain hardly shied away from that in a farewell, posthumous message released on Monday in which he echoed his concession speech to Obama from a decade ago.

"Do not despair of our present difficulties but believe always in the promise and greatness of America, because nothing is inevitable here," McCain wrote in the statement released after his death. "Americans never quit. We never surrender. We never hide from history. We make history."

But as the remarkable story of McCain's life has been replaying this week -- his acts of wartime heroism and his admissions of congressional mistakes -- it's striking the degree to which the old quarrels seem almost charming in the era of Trump.

Burying the hatchet

Not mincing words, as McCain rarely did, one of his friends explained it like this: Trump has made it far easier for McCain to bury the hatchet with Bush and Obama.

It was Obama, in particular, who got under McCain's skin long before they faced off in the general election of the 2008 campaign. Their first skirmish happened two years earlier in the halls of the Senate, where McCain blasted Obama for "posturing" on one of his favorite issues: lobbying reform.

"I understand how important the opportunity to lead your party's effort to exploit this issue must seem to a freshman senator, and I hold no hard feelings over your earlier disingenuousness," McCain wrote in a searing and sarcastic letter to Obama

I was thinking back to that clash this week, which ended three days later as McCain and Obama playfully cocked their fists and mugged for the cameras on Capitol Hill.

Those early dust-ups, of course, grew into full-blown fights on the campaign trail, when McCain aired his long-held observation that Obama was full of himself.

"I don't seek the presidency on the presumption that I am blessed with such personal greatness," McCain said one day in the early stages of their race . "That history has anointed me to save my country in its hour of need."

What annoyed McCain the most, of course, was how Obama was steadily capturing the mantle of change and the air of excitement that he once enjoyed aboard his famous campaign bus, the Straight Talk Express.

I was lucky enough to see that for myself when I first met McCain in the fall of 1999, as he was becoming the rising candidate in the Republican primary. At the time, I was a cub reporter for the Des Moines Register. Since McCain's strategy was to skip the Iowa caucuses -- a very wise decision -- I went to find him in New Hampshire and was overwhelmed by his charm.

Charm turned cantankerous

By the 2008 campaign, after I had interviewed McCain many times while covering Congress for the Chicago Tribune and The New York Times, his charm turned cantankerous. McCain snapped at me one day as he returned to Washington to cast a vote, curtly suggesting reporters were treating Obama with kid gloves. He often refused to speak to journalists, but he always had a hard time keeping the punishment up for long because he did like talking -- and promoting -- issues that mattered.

It was Obama, ironically, who was far more reticent around the press. He initially confided to advisers that he was worried about running against such an authentic figure and war hero like McCain, but quickly found a way to agitate his opponent: tying him to Bush.

He did so again and again as they tangled in televised debates during the fall of 2008 in Mississippi, Tennessee and New York and in campaign rallies in all corners of the country.

"I guess that was John McCain finally giving us a little straight talk and owning up to the fact that he and George Bush actually have a whole lot in common," Obama said in a frequent refrain. "Here's the thing, we know what the Bush-McCain philosophy looks like."

All of that, of course, is ancient -- and quaint -- history. Yet it will be front-and-center on Saturday as Bush and Obama, an improbable duo, rise to send off McCain, a rival no more.