Thanks for some of the excellent suggestions! I may be filling my movie quotient for the next year. (I don’t watch many films.)
I’ll second Primer and Equilibrium.
I’d like to add Tampopo. Think John Wayne meets Iron Chef meets Karate Kid…
However, it can be difficult to find, I believe it was out of print the last time I checked but it’s definitely worth the view if you can find it.
I’m just here to second Once, Dead Man, El Topo, and 1776. El Topo and Dead Man are westerns–my favorite kind of western film, where everything is absolutely fucked up and bind-bendingly absurd, and often hilarious in a very dark way.
I’ve got the soundtrack for 1776 in my car right now. I think it might be one of my favorites ever.
“It doesn’t matter. I won’t be in the history books anyway, only you. Franklin did this and Franklin did that and Franklin did some other damn thing. Franklin smote the ground and out sprang George Washington, fully grown and on his horse. Franklin then electrified him with his miraculous lightning rod and the three of them- Franklin, Washington, and the horse- conducted the entire revolution by themselves.” - John Adams in “1776”
People who like The Dish may also like The Castle, by the same people. A very charming film, and popular in Australia.
That should be mind-bendingly. sigh
Swear to god, it came up on my “saved” list – though of course when I clicked your link, it showed as being in my queue. 
I have wanted to see this movie for 20 years – have never found a copy.
And RealityChuck – thanks for the links, I’ll look both of those over more carefully later – because movies in one’s Netflix queue are like the stacks of books all over my house – it’s impossible to have too many.
Looked again when I was playing with my queue – it is, in vact, in my queue – in the “Saved” section, which generally means they don’t currently have a copy to rent, though they may have in the past.
The Forbidden Zone has to rank up there with Eraserhead as one of the balls-out weirdest films ever made. It’s a bizarre, indescribable fantasy musical featuring Hervé Villechaize, composer Danny Elfman as Satan (he provides the score, too), and the delectable Gisele Lindley, who fortunately spends most of the film topless.
The film was directed and written by Richard Elfman, Danny’s brother, founder of the Mystic Knights of the Oingo-Boingo.
The Forbidden Zone may not be for everyone, but I guarantee that you will never forget it.
I’ll also recommend The 1998 Australian film The Interview (not to be confused with about seven or eight other films of the same title), starring Hugo Weaving (aka Agent Smith). It’s about a man who’s suddenly dragged out of his bed and arrested. The rest of the film takes place at the police station, where we, the viewers, try to figure out who’s the good guy and who’s the bad guy. Fascinating, and brilliantly acted.
Thirded.
Eyebrows of Doom’s “Proof” is a good choice, too, as is commasense’s
“The Interview.”
For my own selection, I will offer Ridicule.
From IMDB:
To get royal backing on a needed drainage project, a poor French lord must learn to play the delicate games of wit at court at Versailles.
The intrigue, the subterfuge, the pettiness, the costumes, the bedroom romps, it’s all there.
I love this movie and watch it once or twice a year, same as Local Hero.
I’ll third 1776 .
If you like movies about kids chasing their show biz dreams, check out Center Stage. It’s about students competing to get into the American Ballet Academy and I’d call it a “nice, little movie” with a fairly catchy soundtrack some interesting choreography and a lot of good dancing. IMDB tells me it also has one of the best ballet dancers in the world, Ethan Stiefel, in it.
Keeping the Faith is worth watching, too. It starts of with a cliche, but not the standard romantic comedy one. A priest and a rabbi don’t walk into a bar; they fall in love with the same girl who was “the coolest girl in the universe” or something like that when they were all 10 years old. Jenna Elfman is the girl; Ed Norton’s the priest; and Ben Stiller is the rabbi. If you get it, get it on DVD. There are two scenes which were cut from the movie which were wonderful. The first one is the funnies bit involving a pager I’ve seen, and the other is touching and adds something to the movie. OK, so it’s a romantic comedy, but it’s not a bad way to spend a couple of hours.
Finally, the Japanese version of Shall We Dance? is very Japanese as in it shows ordinary Japanese life, yet very nice and sweet. The subtitles are well done, and include some particularly funny and accurate translations, including “Your dancing creeps me out!” which is the rudest thing said or done in the movie. The American version with Richard Gere isn’t bad, but, because it isn’t quite as good because of the differences between American and Japanese cultures.
Here are a few of the film’s I’ve talked about on my “Great but Forgotten” page. Most would fit the OP (and a few have been mentioned).
All These Women — Slapstick comedy. By Ingemar Bergman. Really.
**Best Boy **— Heartwarming documentary about a retarded man learning to live life.
Black Moon — Louis Malle’s mysterious film.
Brassed Off — British comedy about music and coal mining.
Brewster McCloud — Robert Altman comedy about a man who tries to fly.
Brute Force — 40s prison break movie with Hume Cronyn as the sadistic guard.
The Body Snatcher — Boris Karloff as the title character, one of film’s great performances.
Comfort and Joy — Quirky gang war comedy set in Glasgow, Scotland.
Cry Uncle! — The raunchiest sex comedy ever.
Cobb — Tommy Lee Jones as one of baseball’s great psychotics.
The Court Jester — “The pellet with the poison’s in the flagon with the dragon.”
Days of Heaven — The most beautiful movie ever made.
Deathwatch — Harvey Keitel in the ultimate in reality TV
The Doberman Gang — Better Than It Has Any Right to Be film about robbing banks using dogs
East Side Story — Fun Documentary about Marxist movie musicals. Really.
**Ella Enchanted **— Delightful postmodern fairy tale with Anne Hathaway
**The 5,000 Fingers of Dr. T ** — Live Action Dr. Seuss from the 50s.
**Flower Drum Song **— Not politically correct any more, but a fine musical.
Force of Evil — Sophisticated 40s film noir about the numbers racket and organized crime.
Hear My Song — Charming British comedy.
Hearts of the West — The wild west was not like in the movies.
The Heartbreak Kid (1972) — A man on his honeymoon has a change of plan.
Hoodwinked — Witty animated retelling of Red Riding Hood.
It Came from Outer Space — Science fiction classic in 3-D.
Jean de Florette/Manon of the Spring — Superb French tragedy.
The Last Waltz — Greatest rock concert film ever.
Little Voice — A shy girl with an amazing voice.
Lost in La Mancha — How a great movie doesn’t get made.
Magic — A young Anthony Hopkins as a ventriloquist who isn’t sure of his sanity.
Max — “What if” film about a Jewish art dealer who befriends a young Adolph Hitler.
Medium Cool — Politics, Protest, and the press in the 60s.
Million Dollar Legs — Wild, absurdist 30s movie comedy.
Movie Movie — Delightful pastiche of 30s movies – and two movies in one!
Mother Night — Fine adaptation of Kurt Vonnegut novel
My Bodyguard — How to deal with a bully – and look who’s in it!
**Neighbors **— The neighbors from hell.
The Onion Field — Realistic police drama, with James Woods at his best.
Picking Up the Pieces — Woody Allen as Tex. Sharon Stone as his wife.
Putney Swope — “Rockin’ the boat’s a drag. You gotta sink the boat!”
Resurrection — Ellen Burstyn as a woman who can work miracles.
Return to Oz — Dark sort-of sequel of the classic – and still pretty good.
Riding Alone for Thousands of Miles — Chinese film about fathers and sons.
Saboteur — Hitchcock classic that was a trial run for North by Northwest.
Scarface — The original classic gangster film by Howard Hawks.
Screen Door Jesus — Faith and humor in a Texas town.
The Smallest Show on Earth — Charming British comedy about running a movie theater.
Stevie — An intriguing look into the life of a smart woman poet.
Support Your Local Sheriff! — Terrific comedy western with James Garner.
The Stunt Man — Peter O’Toole as a mad film director in a great film about moviemaking.
**The Tall Blond Man With One Black Shoe **— Long title, very funny spy spoof.
Targets — Boris Karloff as an aging horror actor facing real horror.
The Tenant — The best horror film ever.
Truly Madly Deeply — A love story from beyond the grave – done right.
200 Motels — Glorious mess from the mind of Frank Zappa.
The Well — A film ahead of its time.
I’m not a great persuader, but you really ought to grab a six-pack of beer, a bag of chips and Cannibals: The Musical (Also known as Alfred Packer, the musical)
Leaving Normal “Thelma and Louise for people with brains.”
Mr. Frost One of the scariest movies I’ve ever seen.
Into the West Not the cowboy show. Set in Ireland (I think) Two little boys and their (stolen, magical?) horse.
His name was Alferd Packer. Weird, I know, but some folks get all het up about it.
And I have just added that to Netflix, thanks. 
Tapeheads - John Cusack, Tim Robbins, Mary Crosby. A rather twisted tale of the early days of the video business. Worth it just for the “B” plot.
*Student Journalist: As your party’s leading presidential candidate, you are a staunch advocate of nuclear proliferation. Does wanting more and bigger missiles stem from a penis envy of the Russians?
Norman Mart: No, son. That’s primitive thinking. I will add, though, that I’ll put my slab on the yard stick against Gorby any day. *
The Odd Angry Shot - Australia’s involvment in Vietnam. By far one of the best movies about the war, and from an oft-unseen angle.
*“Welcome to Patrol 2-2, specialists in arson, murder and drinking.” *
Water - Michael Caine, Valerine Perrine, Billy Connolly. Engaging little farce about an abandoned oil well that strikes Perrier on a small island. Worth it if only for the “Concert for Cascara.”
Baxter Thwaites: How can we maintain decorum if our spiritual leader is trying to put his hand up Miss Cascara’s skirt?
Delores: You prefer he put his hand up my skirt?
Baxter Thwaites: At least you’re used to it. You won’t scream.
Up The Creek - Porky’s/Animal House ripoff featuring cast members of both movies in a college rafting race. Quite funny and witty, really. With boobies!
I heartily concur. FYI, the audio clip played at the beginning of the song Tubthumping comes from this movie. ("I thought it mattered. I thought the music mattered…)
**Kiss Kiss Bang Bang **
Everyone I show this movie to loves it and wonders why they haven’t seen it before.