Interesting suggestions from all! I think everything has a whole 'nother level of depth with headphones, and the contributions thus far definitely reiterated this thought.
Some that seriously made me go

with headphones:
Spoiler for Brace yourself for an ungodly amount of vidjos::
Woods of Desolation - Somehow
Beautiful soundscape, really. The kind of thing I can kind of lay back, close my eyes, and get lost in its melancholy.
Muse - Uprising
I become such a huge proponent of Muse when I'm wearing headphones.
Radiohead - Like Spinning Plates
Aside from being the most depressing song imaginable, this is an absolute treat to listen to with headphones.
Blonde Redhead - Falling Man
This one is just great to listen to.
Sarah Brightman - Anytime, Anywhere
Sarah Brightman has the most lovely voice I've ever heard. With headphones, it's enhanced that much more. I desperately hate the semi-poppy, English language chorus in this one, but the verses are absolutely lovely. This song showcases her voice at its absolute prettiest.
Queen - The Prophet's Song
The toy koto is gorgeous in this one, and the vocal interlude is definitely BEST appreciated with headphones. This is on the same album as "Bohemian Rhapsody" and is really the more intricate and adventurous composition of the two. The way this is arranged is near perfection in headphones. Dat guitar, my God.
Phish - You Enjoy Myself
Best enjoyed with a pair of headphones. As an aside, if I could play this on guitar, I'd feel like a goddess.
Jethro Tull - Aqualung
I love listening to this on headphones like crazy, like whoa.
The Beatles - Yesterday
I don't even like this song without headphones. Some might say the weird stereo panning takes away from the song, but I think it actually adds some depth and interest to what could easily be a hellaciously boring ballad.
Themselves - It's Them
The first time I heard this, it was with good headphones, and I was completely awestruck.
Hawkwind - D-Rider
Just fantastic.
Tuxedomoon - Music Number Two (Reprise)
My ears came the first time I heard this one.
Soft Machine - All White
A wondrous experience through headphones. There's a lot to take in.