Shanksville, Pennsylvania - President Joe Biden donned one of his predecessor and adversary Donald Trump's signature red hats on Wednesday in a move the White House said was a show of "unity" on the anniversary of the September 11, 2001 attacks.

The veteran Democrat was dared to put on the cap as he attended commemorative events with Vice President Kamala Harris – Trump's opponent in the November presidential election – at the Pennsylvania site where Flight 93 crashed.

"At the Shanksville Fire Station, President Joe Biden spoke about the country's bipartisan unity after 9/11 and said we needed to get back to that," White House spokesman Andrew Bates posted on X.

"As a gesture, he gave a hat to a Trump supporter who then said that in the same spirit, [Biden] should put on his Trump cap. He briefly wore it."

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Biden (81), who was replaced by Harris on the Democratic ticket after withdrawing from the race in July, has repeatedly characterized Trump as an existential threat to US democracy.

Video and still images of the president in the hat, bearing the slogan "Trump 2024," quickly went viral.

Several commenters noted that the episode came the day after a prime-time televised presidential debate between Trump and Democratic nominee Kamala Harris, in which the Republican told his opponent: "Biden hates you."

The Trump campaign's War Room rapid reaction account quipped that Biden had put the cap on because "Kamala did so bad in last night's debate."

In reality, Trump is considered by neutral observers to have had a poor night, coming across as defensive, unprepared, and obsessed with the past and personal grievances.

Across three national polls conducted since the showdown on ABC, an average of 57% of viewers said Harris turned in the better debate performance, while only 34% said Trump came out on top.

Earlier on Wednesday, Harris and Trump shook hands at New York's 9/11 memorial.

Wearing commemorative blue ribbons, they watched with other assembled dignitaries, including Biden, as the names of the almost 3,000 victims of the attacks on the Twin Towers were read out.