They are, perhaps, the ultimate odd couple in American politics. Republican and Democrat, white and black, one drama-prone, the other chill.

Of course, they aren’t really a couple. But as the current and immediate past president of the United States, Donald Trump and Barack Obama are joined at the hip by history. President Trump’s performance is being judged, foremost, against that of his predecessor. And in the short period since they first met in person – three days after the election – they have gone through a dizzying ride of ups and downs.

The most bizarre moment of all came Saturday, with Mr. Trump’s eye-popping – and unsubstantiated – tweet-storm alleging that President Obama wiretapped Trump’s office during the 2016 campaign. Mr. Obama and members of his administration have denied the charge.

“Really, the relationship between these two men could fill a book – especially on Trump's end,” writes Aaron Blake in The Washington Post's political blog The Fix. “From its earliest days, Trump seems to have been torn between respect for Obama and a desire to accuse him of high crimes using dubious information.

“Obama's posture toward Trump, meanwhile, has been steadier – a kind of diplomatic tolerance, with the occasional jabs (when he deemed them to be appropriate),” Mr. Blake adds.

Most former presidents lay off their successors, though there are exceptions. More unusual is today’s scenario, with a new president claiming conspiratorial behavior by his predecessor.

“Historically, presidents rarely attack their predecessors,” says historian David Pietrusza. “Woodrow Wilson responded to Theodore Roosevelt's jibes by musing, ‘The only way to treat an adversary like Roosevelt is to gaze at the stars over his head.’”

“Basically, hitting back at an ex-president is punching down, and that's never a sound strategy for any politician,” says Mr. Pietrusza, an author of six books whose primary focus is presidential history. “But Donald Trump doesn't play by normal rules – or, maybe, by any rules at all. So it's nothing for him to punch downward or backward or sideways.”

Blake provides a handy tick-tock of the Trump-Obama “relationship” over the years. An analysis of the timeline suggests that Trump has used Obama as a foil, a character against whom to position himself. Trump tears him down, builds him up, then tears him down again, sometimes in a matter of days or within the same interview.

“Trump’s the kind of guy who can blow hot and cold,” says veteran Democratic strategist Peter Fenn. “One minute he’s giving you a great big hug, the next minute you’re fired.”

Add to the mix the fact that Obama lives and works in Washington, which may be an irritant to Trump. Obama has maintained a low public profile, but has put out statements on a few occasions – to register his objection to Trump’s first “travel ban,” and then to deny that he had wiretapped Trump.

Contrasting job approvals

During his final stretch in office, Obama enjoyed high marks from the public, departing the White House with a 57 percent job approval. Trump, in contrast, took office with historically low ratings, which remain in the mid-40s. Perhaps, in opining a month ago that Obama liked him, Trump was hoping to improve his own image.

But that period of professed good feeling didn’t last long. By the end of February, Trump was accusing Obama of fueling protests against the new president. And now, Trump is charging Obama with engaging in a “Nixon/Watergate”-level conspiracy against him. On Monday, White House press secretary Sean Spicer asserted “substantial reporting” to warrant congressional investigation of the alleged wiretapping, but did not offer specifics.

The timeline of the Trump-Obama relationship depicts two larger-than-life men. Though Trump was largely complimentary of Obama during the latter's days as a likely presidential hopeful and into his early presidency, that all ended in March 2011 when Trump began to push “birtherism,” the false claim that Obama may not have been born in the United States. Probably not coincidentally, Trump was thinking of running for president in 2012.

When Obama released his long-form birth certificate in April 2011, Trump did not relent, and only acknowledged Obama’s American birth last September.

Obama, too, has taken his shots at Trump over the years. In 2008, he called out Trump as a wealthy person who can file for bankruptcy to fix his finances. At the 2011 White House Correspondents’ Dinner, Obama mocked Trump as the billionaire/reality-TV star sat in the audience.

“Now I think he wonders whether he did the right thing, picking on him at the correspondents’ dinner,” says Mr. Fenn.

In the heat of the 2016 campaign, Obama was unsparing in his disdain for Trump, calling him “unfit” for the presidency. Likewise, Trump slammed Obama repeatedly as a “failure.”

An exclusive club

Post-election, the gloves went back on, and Obama assured Trump a smooth transition. After their first-ever meeting, a 90-minute session three days after the election, Trump called Obama “a very good man” and spoke of the “chemistry” between them. Obama told Trump, “If you succeed, the country succeeds.”

Such sentiments are typical of the exclusive club known as the American presidency. Indeed, the two Presidents Bush were famously cordial to their successors; Bush Senior and President Clinton developed an especially warm friendship.

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Since the election, Trump and Obama have spoken numerous times, according to Trump aides. There were good reasons for Obama to maintain an open line of communication to his successor: A smooth transition would reflect well on Obama, particularly in handing over the keys to a man who had never held public office. It’s also in Obama’s interest for Trump to understand the rationale for his policies, and encourage continuity. On foreign policy, in particular, Trump has fallen in line with many Obama positions. In the domestic arena, Trump has expressed interest in keeping key elements of the Affordable Care Act even as he transitions away from the law.

Now all bets are off. Trump’s policy choices are one thing, but any kind of good working relationship between Trump and Obama seems off the table. And where the wiretapping allegations go is anyone’s guess.