Do sawtooth waves that ramp upward and sharply drop sound
the same as sawtooth waves that ramp downward and sharply
rise (reverse sawtooth waves)? If not, then how different do
they sound, and in what way do the sounds differ?

If we're talking inversion (rotating upward/downward by a 180 degrees) then no. The difference is not audible.
However, if you mix the two (original and inverted together) they will cancel each other out.

I'm too tired to go on. Morph should take over.

Good night,

EDIT:
Well, I'm back.
Indeed, as mentioned below if you use one as a modulation source the difference can be quite dramatic. unFortunately only a few synths allow for this.
-ED-

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A gentleman is one who never hurts anyone's feelings unintentionally.
- - - Oscar Wilde

[This message has been edited by 3351 (edited 02-14-2006).]

Waveforms that have the same harmonic (overtone) structure sound the same, which is true for the case that you mention. However, that assumes the sound is being heard directly, and not the result of using it to modulate or otherwise interact with another source. For example, the ramp could be used as an LFO, and depending on what it modulated/controlled, the resulting output would sound quite different depending on the slope of the rise and fall.

--Barry