Mr. Curly Twirly entered Mopatop's Shop in the episode "Wind", just after Puppyduck let the wind out of its container. But before he can say what he's looking for, the wind blows his hat off his head. Puppyduck helps him catch the hat and he decides that he will come back after they have captured the wind.
His name is not mentioned in the episode, but specified in the credits.
The puppet had previously appeared as an unnamed background character, beginning in 1992, with The Muppet Christmas Carol, followed by Muppet Treasure Island, Muppets Tonight (in the audience), and The Wubbulous World of Dr. Seuss (as a Fiff-a-ma-toze in "There Is Nothing To Fear In Here"). The monster also appeared on the cover of the 1993 book Kermit Learns How Computers Work.