There are more complicated things like cones. (Open sets in) Weyl chambers might be present, for example. One thing that I know, is that the orbit space of a finite groups action is stratified into submanifolds which correspond to orbit types corresponding to cojugacy classes of isotropy groups inside the finite group. There is always one big connected and locally connected open dense stratum, the regular stratum. I can point you to sections 29 and 30 of here. Maybe, also this paper is of interest to you (there are some mistakes in it). And this other paper is devoted to Riemannian geometry of orbit spaces of isometric group actions in general.

There are more complicated things like cones. On the other hand, since the action is linear, each orbit spce has an action of $\mathbb R_{>0}$ (is a cone). (Open sets in) Weyl chambers might be present, for example. One thing that I know, is that the orbit space of a finite groups action is stratified into submanifolds which correspond to orbit types corresponding to cojugacy classes of isotropy groups inside the finite group. There is always one big connected and locally connected open dense stratum, the regular stratum. Besides this, codimension 1 strata (walls) can only come from reflections, if I remember correctly. For this I can point you to sections 29 and 30 of

• Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008. (pdf).

And the following paper is devoted to Riemannian geometry of orbit spaces of isometric group actions in general:

• Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: The Riemannian geometry of orbit spaces - the metric, geodesics, and integrable systems. Publ. Math. Debrecen 62 (2003), 247-276. (pdf)

Maybe, also is of interest to you (there are some well hidden mistakes in it).

• Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Reflection groups on Riemannian manifolds. Annali di Matematica 186 (2006), 25-58. (pdf)