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.