  We study the symplectic reduction of the phase space describing $k$ particles
in $\mathbb{R}^n$ with total angular momentum zero. This corresponds to the
singular symplectic quotient associated to the diagonal action of
$\operatorname{O}_n$ on $k$ copies of $\mathbb{C}^n$ at the zero value of the
homogeneous quadratic moment map. We give a description of the ideal of
relations of the ring of regular functions of the symplectic quotient. Using
this description, we demonstrate $\mathbb{Z}^+$-graded regular
symplectomorphisms among the $\operatorname{O}_n$- and
$\operatorname{SO}_n$-symplectic quotients and determine which of these
quotients are graded regularly symplectomorphic to linear symplectic orbifolds.
We demonstrate that when $n \leq k$, the zero fibre of the moment map has
rational singularities and hence is normal and Cohen-Macaulay. We also
demonstrate that for small values of $k$, the ring of regular functions on the
symplectic quotient is graded Gorenstein.
