No, it wasn’t A.I. But it was no endorsement, either.

A smiling President Biden really did wear a Trump hat — for about five seconds — while visiting firefighters for an event commemorating the Sept. 11 attacks.

The unlikely moment, which played out on Wednesday at the Shanksville Volunteer Fire Department in a Republican-leaning area of rural western Pennsylvania, soon turned into another example of America’s split-screen political culture.

It started as Mr. Biden exchanged friendly banter with a man wearing a red “Trump 2024” hat, offering him a blue cap with the presidential seal to put on instead. The man asked if the president would autograph the blue hat.

“Sure, I’ll autograph it,” Mr. Biden said. The man replied, “You remember your name?”

“I don’t remember my name,” the president joked. “I’m slow.”

The man teased, “You’re an old fart,” and Mr. Biden immediately agreed: “Yeah, I know, man, I’m an old guy.”

Then, after signing the blue hat, Mr. Biden told him, “I need that hat.”

The man took off his red Trump cap and handed it over, and Mr. Biden, at the urging of the crowd, put it on his head, over a baseball hat he was already wearing. The crowd — including the smiling and clapping Trump supporter — let out a cheer as the president grinned, displayed the hat to the firehouse and then took it off and shook the man’s hand.