Thanks for the reply! I'm not sure I fully got that, though. So it seems to be that it's not actually about position (the absolute coordinates), but about the velocity of the particle? So, you could just always use a coordinate system that has the particle at its origin so that its position doesn't need to change, and just invert the vector of its velocity to get the same result?

Edit: Went over the Wikipedia article, I think that cleared it up a bit - it's not actually about a single particle being inverted in an otherwise unchanged system, but the whole system that you're observing being inverted, is that correct? In that case, it would actually not matter what point is chosen as the origin, as the relative positions of everything would work out to be the same no matter the origin of the inversion. That makes a bit more sense then.