I'm not sure I would characterize this piece as C major at all, despite the fact that it is written with no key signature, and ends with a C in the bass. It never really "toniciszes" the C with a dominant/tonic relation (nor is it in A minor, for that matter). In fact, I believe it is, in many ways, more akin to atonal music.
In addition to the 5/4 meter that Tim mentioned, the piece makes heavy use of dissonances -- especially minor seconds -- throughout, and it often appears to be in multiple keys at the same time (polytonality). For example, at 1:20, where the bass finally resolves to a C (from the G that it had been droning up to that point), the chord in the rest of the instruments is actually a D♭! (A half step above the C, and a tritone above the G). Using tones a minor second above the bass note recalls the darkness of the Phrygian mode.
Even in the the spots were major chords are used prominantly, they are used in such a way that they would never occur in any single key. For example, there's a passage (around the 4:00 mark) of completely parallel major chords that goes: Cmaj, Bmaj, B♭maj, Bmaj, Cmaj, Dmaj, Emaj, F♯maj, G♯maj, Gmaj (all set against a B drone). Not only do all the notes of each chord not belong to a single key, but even the roots themselves don't all belong to a single key! That's just one example bar, but there are many similar (but slightly different) bars throughout the piece.
There's really never any sense of "this is the tonic, this is home" throughout the piece, which, as I mentioned, is achieved by avoiding dominants and using heavy dissonances, and notes from multiple keys.