For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$  In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$.  For two elements $g,g'\in\mathrm{SO}(3)$, define an equivalence relation $\sim$ via rotations of the axis-aligned cube:  $g\sim g'\iff (g\cdot [0,1]^3 = g'\cdot [0,1]^3)$.

**What is the structure of $\mathrm{SO}(3)/\sim$?  Does this space admit a simple parameterization, measure, or metric?**

More broadly, is this a well-studied space with a name?  Some interesting problems in discrete geometry seem to involve this structure.

*Note:*  $\mathrm{SO}(3)$ is a simple group, so I expect this to a quotient space rather than a quotient group.  This is a refined version of an [earlier question][1].


  [1]: http://math.stackexchange.com/questions/1403374/space-of-arbitrary-rotations-of-a-cube