Broken Base: There's no real unanimous agreement over which is the best way to see the film: with the Japanese version, the Troma/Fox/Streamline dub, or the Disney English dub? As with most hotly debated dubs, you'll have to decide for yourself.
The visual◊ of Satsuki with Mei on her back waiting at the bus stop with Totoro is often redone with characters from other series by fan artists.
Miyazaki's Syndrome, which first appeared in this movie—during the scene Mei plays with the sleeping Totoro rather than displaying the slightest bit of fear or even hesitation. That Dude in the Suede describes it as "all children losing every sense of self-preservation and only experiencing childlike wonder and amazement."
Retroactive Recognition: Elle Fanning voiced Mei in the Disney dub, along with her sister as Satsuki. A few years later, she would go on to play Aurora in 2014's Maleficent (though she also had other roles between that film and My Neighbor Totoro).
You will not believe how some people jump to the conclusion in the scene where the father and his daughters are bathing together that it must be pedophilia when that situation is considered perfectly normal and wholesome family time in Japan and many other countries. Including some parts of the US, actually (although usually when the children are very young, i.e, not Satsuki's age).
The sequence where Mei gets lost attempting to deliver an ear of corn to her mother includes her sitting by a row of statues. In Japan, such statues represent the Bodhisattva Jizo, the Buddhist deity who is the protector of children, so the director was trying to convey to the audience that Mei is in no immediate danger while her sister and King Totoro come for her in the Cat Bus. (Ironically, within That Other Theory, this scene is offered as one of the proofs that Mei is dead, since Jizo is also the protector of dead and unborn children and infants.)
There is a scene in the middle of the movie where Mei and Satsuki are helping clean tatami mats. It's an unremarkable scene for anyone from Japan (or anyone with a knowledge of Japanese culture) but viewers from other regions were often left confused as to what the two girls were doing.
When the film was first translated to English, Studio Ghibli negotiated full editorial rights (after a Macekre of Nausicaš of the Valley of the Wind) over the translated versions and indicated that the film was to be faithfully translated and shown with no censorship, alterations, or script changes. The translators nonetheless requested the removal of the Furo Scene and the tatami-cleaning scene, citing Values Dissonance as their reason. Ghibli's response was to send them a katana with a note attached saying "No cuts!"