It’s nice when a boss can have a sense of humor, especially when he is the president of the United States.

George Bush understood that.

Long before Mr. Bush flaunted his personality with eye-catching socks or celebrated milestone birthdays by sky diving, he showed off his lighter side when he secretly invited Dana Carvey, the comedian who impersonated him on “Saturday Night Live,” to take the stage at the White House.

The exchange, which was captured on video, is an endearing moment in what became a yearslong friendship between Mr. Bush, who died on Friday at 94, and the man who parodied him.

It was December 1992, and Mr. Bush, who had been defeated by Bill Clinton, was on his way out of the White House. He summoned his staff to the East Room for a formal Christmas greeting. But when “Hail to the Chief” began to play, it was not Mr. Bush who entered the room, but Mr. Carvey.