Let $G\to GL\left(V\right)$ be a representation of a reductive linear algebraic group $G$ on a finite-dimensional vector space $V$, defined over an algebraically closed field of characteristic zero. The categorical quotient $X=V//G$ carries a natural stratification, due to D. Luna. This paper addresses the following questions:(i) Is the Luna stratification of $X$ intrinsic? That is, does every automorphism of $V//G$ map each stratum to another stratum?(ii) Are the individual Luna strata in $X$ intrinsic? That is, does every automorphism...