1) ALL overdrives compress. Every last stinking one of them. The reason I can say that is because they generate harmonic content when they run out of headroom, and it is the additional harmonic content we seek from them. The question is how much compression they introduce in producing the harmonic content they do. Some introduce more than others.
2) The general strategy for maintaining as much dynamic range as possible whilst still getting the additional harmonic content, is to a) keep a fairly high clipping threshold so that there is a bigger range of input signal amplitudes between the noise floor and the point where clipping sets in, and b) distributing clipping across multiple stages so that each stage is not imposed on so much as to exhaust its headroom and produce compression.
3) Many people will tout the virtues of "asymmetrical clipping" in their descriptions, goals, or ad copy. In truth, the clipping produce by diodes is rarely asymmetrical in the purist sense (i.e., all one set of harmonics and none of the other). I could make a pedal with 2 LEDs in one direction and 4 in the other for "asymmetrical clipping" and I probably wouldn't hear any clipping at all; it's all in the relation to signal amplitude and power supply, not JUST the diodes. But what differential dfiode clipping of different half-cycles WILL do is raise the clipping threshold enough for one half cycle that you get the additional dynamics, while keep the threshold lower for the other half cycle so that you still get to hear a reasonable amount of clipping. That is, i fact, what a lot of folks like about the varios pedals that aim for asymmetry, like the Boss SD-1, or many of the Fulltone pedals, or even the Interfax Harmonic Percolator.