Not a mathematician, and the question probably does not make sense but asking it anyway:
In usual euclidian spaces, central inversion through O(0,...,0) of point M(x1,...,xN) gives you one and only one point M'(-x1,...,-xN). I would write this $$I_O(M)=M'$$ Do you know of any bizarre space where this wouldn't be the case ? Where : $$I_O(M)=M' \text{or } M''$$