In the following article of M.A.Mikhailova (М.А. Михайлова)
Изв. АН СССР. Сер. матем., 48:1 (1984)
О ФАКТОРПРОСТРАНСТВЕ ПО ДЕЙСТВИЮ КОНЕЧНОЙ ГРУППЫ, ПОРОЖДЕННОЙ ПСЕВДООТРАЖЕНИЯМИ.
http://www.mathnet.ru/links/33220b8c84645bec685e85bf17d65994/im1420.pdf
it is proven:
Theorem. The quotient $\mathbb R^n/G$ by a linear action of a finite group $G$ is homeomorfic to $\mathbb R^n$ if and only if $G$ is generated by pseudo-reflections (i.e, rotations of $\mathbb R^n$ that fix a subspace of codimension 2).
The proof relies on a complete classification of finite groups generated by pseudo-reflections (there is a reference to this classification at the end of the article)
(there should be of course an English translation of this article, but I can not find it now).