Maybe this is a stupid question... But why restrict the relative configuration space to those states that obey euclidian 3-geometry or whatever? ie, shouldn't the physics be such that that is a consequence? The fact that "from the inside" it looks like a euclidian 3-space or whatever is more an illusion... it's not the reality, the configuration space is the reality. So why would it "know" to obey the triangle inequality?

Besides, by relaxing that (and the various other restrictions due to the demands of the inside looking like 3-space) one could more easily represent curvature, right?

Oh, may be a stupid thought, but as far as parity: let the values go negative... with a sign flip corresponding to a parity flip, maybe? Actually, not sure about this... then two negative coordinates would have to act equivalent to no negative coordinates, I think.