"I am among other things using it for shape matching. I don't necessarily want to avoid a method based on a rest pose, I have already implemented this. It's just that I am almost completely sure it can be done without it. If a particle cloud's linear state, moment of inertia, angular momentum, velocity, and energy can be computed accurately from particle state, then so can the angle. I just haven't figured out how yet..."
I'm not an expert but I don't think that your conclusion necessarily follows.
It just doesn't make sense to me: particles only have a defined position, not an orientation. You can aggregate their positions to get a COM, which helps you define all the other properties you listed (eg angular momentum doesn't make sense for a particle in isolation, but does relative to the COM), but AFAICT the COM doesn't help define an orientation.
I'm struggling to see how you could define orientation without some sort of frame of reference similar to that provided by a rest pose.