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.