The name of this episode is a reference to the Children's book The Cat in the Hat by Dr. Seuss.
Homer travels into the South Park universe in the couch gag.
The song that plays when Lisa is with the hat is Wouldn't It Be Nice by The Beach Boys.
Bart says that he is bad to the bone, a reference to the George Thorogood song "Bad to the Bone".
Donald Trump appears in Lisa's fantasy and he appears to be tweeting on Twitter.
Rod and Todd's nosebleed after levitating the compacted car through faith is a reference to Stranger Things. The character of Eleven in the series has a psychokinetic power which leads to a nosebleed after each use.
Mozart's Requiem in D Minor makes several appeareances while Homer reminisces about his past on chess.
Bart's guilty conscience resembles his non-canon twin brother Hugo Simpson II.
A display of Bort license plates is seen in the background as Bart takes his "Bad to the Bone" tattoo, which is a reference to the episode "Itchy & Scratchy Land".
The episode is framed as a story being told by Bart and Lisa, but they weren't present during most of Homer's B-plot. How did that get into the story?