Okay, if you want to go with strange movies that weren’t intentionally strange, I’m nominated The Room.
I once showed this at one of my yearly Bad Film Festivals (THe theme that year was “Failed Cult Movies”).
Only partway through the opening credits, my wife begged me to turn it off. She was outvoted, but five minutes later, everyone else voted to turn it off.
Aside from people on this Board, I’m the only one I know who has watched it all the way through.
Yet the first time I saw it was on HBO. And they had it prominently available for rental at the video rental place I used for years. I think a lot of people saw it and don’t want to admit it.
It’s Whoopi Goldberg in The Telephone.
The Greasy Strangler.
About someone who likes to grease himself up and strangle people until their eyes pop. Then go and wash it off down the local automatic carwash which is run by a blind man.
Especially the two minute sequence based around the pronounciation of the word Potato.
Mandy with Nick Cage has to be on this list
Holy Motors was supposed to be mainstream, wasn’t it? It certainly did ok at film festivals.
I love weird movies including all the Terry Gilliam ones mentioned. The mother of weird movies for me has to be Darren Aronofsky’s “Mother!” with Jennifer Lawrence. It starts off kinda quirky, but man, when it gets rolling, it’s hard to imagine filming anything weirder. I can’t say I enjoyed it, but it’s definitely memorable.
For mainstream, after watching Even Cowgirls Get the Blues I was left wondering “what the fuck did I just watch?”. I still couldn’t describe the plot to this day, other than Uma Thurman had huge thumbs.
Most of the good(?) ones have already been mentioned, so I’ll go with Gentlemen Broncos, which featured(?) one of the most bizarre projectile vomiting scenes I’ve ever witnessed.
I’ve seen “Pink Flamingos”. Need I say more? 
The weirdest “documentary” I’ve ever seen has to be a low-budget doco I got from Netflix called “Patient Seventeen”. It was ostensibly about a podiatrist (who died before filming completed) who was willing to remove alien implants from people’s feet and lower legs, and then it veered off the rails and they interviewed some people who seemed shady in ways I couldn’t seem to define. I Googled one of them, and he turned out to be one of the people who was interviewed as John Doe #2 in the Oklahoma City bombing, and still hasn’t been fully exonerated.
Other people on this board have talked about a low-budget movie called “Gummo” that is loosely based on the tornado that nearly destroyed Xenia, Ohio. That was really weird, too.
My local PBS station aired a cut-for-length movie a while back called “Rat Film” that was a total mind-fart, and I shut it off halfway through. I thought it had critical parts scissored out and that was why I couldn’t understand it, but IMDB said otherwise. Apparently, it won some awards at SXSW and that’s how it landed on PBS.
My Brother’s Keeper, a documentary with the tagline, “A Heartwarming Tale of Murder”. It really made one think about life, death, justice, morality etc.
Donny Darko. My brain is still mushy from those 2 hours.
I’ve not seen Forbidden Zone, but a movie mindfuck with a similar name is The Forbidden Room.
I once dated a man who, when the relationship started to deteriorate, said, “I should have known, when you told me you liked ‘Eraserhead’.” :o Anyway, he and I watched “Kentucky Fried Movie” together on Betamax, and when I was in college a few years later, I found it on VHS and I watched it with my BFF. Neither of us thought it was all that funny, this time around, and when my roommate came home after closing down a bar, put it in the VCR and while watching it, thought, “I cannot believe NWH would watch something like this.”
TCM sometimes shows really bad modern movies, and one they show a couple times a year is called “Heavenly Bodies.” Len Maltin rated it “BOMB” and said, “The first Aerobics Musical, and with any luck, the last.” It’s an unintentional comedy.
“Koyaanisqatsi” is absolutely brilliant, but you have to be able to watch it from start to finish without a break to really appreciate it. “Powaqqatsi” is nowhere near as good, and I haven’t seen “Nagoyqatsi.”
I saw it on VHS not long after it came out, and later on DVD when that became available. I listened to the commentary track, and the filmmakers did say at the end what they thought really happened.
I personally disagreed; I will explain if anyone PMs me.
If you liked Koyaanisqatsi then I would recommend Samsara. IMO, it is better than Powaqqatsi and Nagoyqatsi.
Mainstream (ish): Sir Henry At Rawlinson End. Hard to describe - a brain-dump of Vivian Stanshall gives you some idea.
Small, Arty etc: I Was Catherine the Great’s Stable Boy, a 5 minute short which is, well, pretty much what it says on the can. The weirdest thing about it is how anyone thought that making it was a good idea.
j
No, not the American one. The Japanese one. I had to watch it twice b/c the first time through I kept saying to myself “I can’t believe this is actually happening.”
(The link has a trailer right there. Tell me that ain’t some weird shit.)
That’s where I was first made aware of a little weird one called Nothing Lasts Forever. I’m not sure what they were smoking when they wrote it, but they should have switched dealers. It’s not as strange as some of the others listed, but it’s up there.
For mainstream movies, “Buckaroo Banzai” was pretty far off the wall.
I also accidentally saw Synecdoche New York. It’s very bizarre, got very good ratings, and I disliked it rather a lot.
Some of the weird movies I’ve watched recently:
Synecdoche, New York
Brazil
Timecode
Memento
Of these, Memento is the one I’d recommend. I like time travel movies, but this is in a sense opposite to time travel! And, instead of science fiction, experts say it presents anterograde amnesia realistically.