One of my favorites: Danse Macabre, by Saint-Saens.
Give me some more!
Gymnopedie by Eric Satie.
It makes me feel all squidgy. I really can’t say why because nothing else I’ve heard makes me feel like that. Moonlight Sonata comes close though.
Holst’s “The Planets”. I saw the Baltimore Symphony perform this long, long ago, and they extended the “ahhhs” on the last piece (“Neptune”) for better than 5 minutes. It was awesome.
Fanfare for the Common Man, by Copeland.
Great choices so far! Thanks.
The Symphonie Fantastique, by Berlioz. Particularly the final movement, with the Dies Irae theme and the church bells.
[QUOTE=Wikipedia]
Leonard Bernstein described the symphony as the first musical expedition into psychedelia because of its hallucinatory and dream-like nature, and because history suggests Berlioz composed at least a portion of it under the influence of opium. According to Bernstein, ‘Berlioz tells it like it is. You take a trip, you wind up screaming at your own funeral.’
[/QUOTE]
Debussy’s The Golliwog’s Cakewalk. It was a classical composer’s take on ragtime (yes, the title is racist, but it’s still commonly played).
There’s also Humoresqe by Dvorak.
Sibelius, Karelia Suite
Arvo Pärt Spiegel Im Spiegel
Carl Jenkins, Songs Of Sanctuary
ETA:
Elgar, Nimrod
I’m gonna throw a couple out there. I don’t know how unusual they are, but seem like they might work:
Funeral March for a Parrot by Charles Valentin Alkan. Here’s a YouTube version.
Raymond Lewenthal edited a collection of Alkan’s weird stuff, “The Grotesqueries of Alkan.” Brilliant. It came with a little mini-record of music appreciation lectures, analyzing Alkan’s style and comparing his work to Mahler’s.
wonderful stuff!
Berlioz “Harold in Italy” is intriguing. The bandit’s orgy is lots of fun!
Mahler’s “Songs of a Wayfarer” is also lovely, and slightly off-kilter.
Saint-Saens’ Third Symphony, the “Organ Symphony” is mostly fairly standard, but the really sharp SURPRISE when the organ kicks in is really something special, and earns it a place in the constellation of the weird.
Fixed links.
Oh, speaking of Danse Macabre: see if you can find the original choral version of “Night on Bald Mountain” by Mussorgsky, the version from the opera “The Fair at Sorochinsk.”
If you’re used to the standard orchestral version…get ready for one hell of an eerie thrill. The choral version really summons up the devil! It’s scary in broad daylight!
(The choral version of Strauss “The Beautiful Blue Danube” is also intriguing. (Here’s the YouTube link. But…ah…it doesn’t really hit it until about the 1:30 mark. Before that, it’s just warm-up.)
It’s fun working your way through Kickass Classical Top 100 - Classical Music Best Famous Popular (and other videos like it at YouTube) putting names to some of those tunes you recognize when you hear them. A good way to kill some minutes, anyway.
I was also going to suggest Mussorsky. That choral version is wonderful - also look for Emerson Lake and Palmer’s electronic rock version, and piano renditions.
Also seeThreepenny Opera. this is NOT your standard opera fare! Search youtube for Polly’s Song or Dreigroschenfinale. (No, really.)
If this means that posting really famous pieces is OK, I might as well link to Stravinsky’s Rite of Spring.
The first time you hear it, you hate it. Then your brain rearranges itself in a new configuration to make sense of it. After that, it becomes awesome. 
Ooh, yes! Damn jolly fun!
Here’s a YouTube link of someone singing “Pirate Jenny.” Not actually a very good rendition, but it gives a hint.
(Are you also a fan of The Rise and Fall of the City of Mahagonny? I just love that to madness! And I adore “The Ballad of the Pirates.” Brecht is like pepper: a little goes a long way!)
Here’s an unusual combination: two pieces, very similar (to my hearing), but neither of which I’d say I like better:
Entrance of the Little Fauves by Gabriel Pierne
and Masks by Prokofiev.
Not too unusual, but one of the few songs that gives me serious chills: Barber’s Adagio for Strings.
When I discovered that the theme song from the old Alfred Hitchcock Presents show is called Funeral March for A Marionette by Charles Gounod, it changed the movie that plays in my mind when I hear it. Before I pictured a frumpy old man tottering around a big old house. Now, I picture stringed puppets dancing jerkily.
This version on YouTubeactually produced animated puppets to accompany the song.
Charles Ives’ America Variations – Ives takes several famous pieces of music and – how else to describe trhis – makes them sound like “America”. Fascinating piece.
Walter/Wendy Carlos did something similar with Pompous Circumstances, where s/he took more recent pieces and forced them into the mold of Elgar’s “Pomp and cIrcumstances”, finishing up with the first minute and a half of Strauss’ Also Sprach Zarathustra.