Consider this: imagining your description gave me goosebumps.
You know, there are times when I want to kick Bach in the nuts. OK, Johann, we know you’re an alien with a musical brain the size of a planet, but do you really have to be obnoxious about it? 
(Of course, then he shows me a bit of this, and I go right back to slobbering all over his privates, and promise that I’ll never be mean to him again.)
Not quite “classical,” but lots of movie soundtrack music: Bernard Hermann’s PSYCHO or VERTIGO, Elmer Bernstein’s THE TEN COMMANDMENTS, etc.
Prokofiev’s Troika: - YouTube
Mozart’s Fourth Horn Concerto: Mozart - Horn Concerto No. 4 in E flat, K. 495 [complete] - YouTube
Bach’s Jesu, Joy of Man’s Desiring: Jesu, Joy Of Man's Desiring (Instrumental) - YouTube
Dvorak’s New World Symphony: Antonin Dvorak - New World Symphony (Full) - YouTube
Saint Saens Le Petit Moulin: Camille Saint Saens Le Petite Moulin - YouTube
Watkin’s Ale: - YouTube
Fantasia on British Sea Songs: Wood - Fantasia on British Sea-Songs / Rule Britannia (Last Night of the Proms 2012) - YouTube
Mike Oldfield’s Tubular Bells, most famous from The Exorcist. One of the greatest feats of composition of the ages.
There’s always the Toy Symphony, by Handel, or by Leopold Mozart, or by Benedictine monk Edmund Angerer, or maybe by someone else. It’s popular fort orchestra fund-raisers, because big donors can get to “play” in the orchestra when this piece is performed, playing the various toys. No musical skill required.
Along those lines, there’s Funeral March for a Marionette by Charles Gounod. It was supposed to be the first part of a longer composition, but he stopped after this first movement. The piece was used in a 1931 radio drama, and as the accompanying music for a 1927 movie, Sunrise, which was seen by a young Alfred Hitchcock, who later used the music as the theme for his TV show Alfred Hitchcock Presents.
Even more so when you know (as I’m sure you do) he played 98% of the instruments himself, overdubbing like a mad bastard, and had to start from scratch when the master tape wore clean through. Jeez!
On the more serious side of unusual instruments, there are a number of works for “prepared” piano, including this one - Mount Hood by Hauschka.
And one more: Zoe Keating’s “layered cello”, in which she employs a Stradivarius cello, a control pedal, and a laptop to record and loop herself as she plays live. “Sun Will Set” was the first piece of hers I heard, and still my favorite.
Alban Berg’s Violin Concerto. It’s a beautiful, emotional piece of atonal music, and it will show you (well, it showed me) that atonal music can be beautiful and emotional.
Same with his opera, Wozzeck, which I had the pleasure of seeing with Deborah Voigt at the Met last season. Berg used the vocal techniques of Sprechgesang and Sprechstimme (sort of pitched speaking, rather than actual singing), and Voight, usually known as (and deservedly famed as) a Wagnerian soprano, handled them magnificently.
(Flanders and Swann. Michael Flanders sings the horn part to Donald Swann’s piano accompaniment. A transport of delight – no, wait, that’s another of theirs. This one is called “An Ill Wind.” The old classical radio station here in San Diego used to play the two together for a hearty one-two punch.)
Charlotte Moorman’s 1960s cello explorations pushed the boundaries of the instrument farther than anyone else ever has. She may have given her life for her art: there is suspicion that she contracted the breast cancer she died of from wearing a bra made of 2 miniature televisions, which Nam June Paik fitted her out with. She got arrested a number of times for performing topless or nude. She played chocolate cellos and ice cellos as well as TV cellos and all kinds of freaky stuff. She was one of a kind.
“Musikalisches Würfelspiel” die-generated compositions certainly would count as “unusual.”
Professor David Cope has come up with some interesting compositions…well, the software he wrote came up with some, anyway. I’ve always been partial to It’s “Another Rag, After Joplin”, mimicking, naturally the style of Joplin. There was also a piece that synthesized the styles of Mozart, and a Balinese Gamelan.
The works of swordsman and composer Joseph Bologne, Chevalier de Saint-Georges, aka “The Black Mozart,” might be of at least unusual historical interest.
Two by Leopold Mozart. (Wolfgang’s papa.)
The Toy Symphony (for many years incorrectly attributed to Haydn.) The orchestra backs up a collection of — toys! Bird calls and rattles and the like. Charming!
The Peasant Wedding. The members of the orchestra are instructed to holler “Yahoo!” and to whistle between their teeth. At a couple of points, a pistol is let off. (And bagpipes and hurdy-gurdy!)
Haydn’s “Missa Sanctae Caeciliae”
Two things I find awesomely hypnotic, I listen to them once in a while and hear them in my head for days, are:
Carl Orff’s Musica Poetica
and Philip Glass’s Prelude to Akhnaten which is built around one musical phrase repeated over and over with different emphasis.
Look up to my post #46. There’s now a belief that Leopold Mozart didn’t compose it, either, but a Benedictine monk I’d never heard of.
I think we could just nominate all of Charles Ives’ production. Nevertheless, I think the companion piece to the Unanswered Question needs to be mentioned: Central Park in the Dark. This is about as close as it comes to a painting in music. It’s incredible that a piece like this was composed in 1906.
From a completely different era, but in a similar genre, 16th century composer Clément Janequin’s Voulez ouyr les cris de Paris (Hear the Cries of Paris). After a short intro in which the title is repeated, we’re presented with a rather cacophonous collection of genuine market criers’ calls. Again, it’s incredible that a piece like this (turnip! turnip!! turnip!!!) was composed 500 years ago. It’s an incredible historical document.
Looking even further back, there’s also the work of 13th century composer Pérotin. He was the most famous member of the Notre Dame School. This was the beginning of polyphony, and it’s music that sounds like it might have been composed ten years ago.
In a completely different genre, there’s W.A. Mozart’s Musikalisches Würfelspiel. Doesn’t sound so weird? That’s true, but it’s nevertheless unusual in that the score is created before each performance by rolling dices and following Mozart’s instructions about how to randomly arrange fragments. It’s John Cage 200 years before John Cage.
Speaking of John Cage, I think his In a Landscape and Dream, both from 1948, are unusual. To our ears, they don’t sound weird, but no one else was writing music like this in 1948. Certainly not “serious” composers.
There are several pieces like that, that aren’t unusual by themselves, but are nevertheless highly unexpected when you consider the composer. Several years ago, all I knew of Stravinsky was his Rite of Spring. I was so shocked when I first put on a CD of his Pulcinella that I had to take the disk out to make sure I didn’t make a mistake. A few people have linked to Legeti’s excellent choral works. He also wrote very different vocal pieces based on Hungarian folk music. Similarly, Japanese composer Toru Takemitsu wrote beautiful and slightly dissonant impressionistic music that blended western and Japanese influences. Serious stuff. I picked up a CD of his title “Song of Circles and Triangles” and expected… not what I got. He also wrote guitar transcriptions of Beatles songs.
A little-known composer that deserves a little bit more recognition is André Caplet. He famously beat out Ravel in the prestigious Prix de Rome competition and was a collaborator of Debussy. Unfortunately, he was not very prolific and died young. He was a master orchestrator and his music features unusual instrument combinations, which is why I mention him here. The Mask of Red Death for harp and string quartet (1908).