Parameterizing rotations of a cube 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:  $g\sim g'\iff (g\cdot [-1,1]^3 = g'\cdot [-1,1]^3)$.
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.
 A: $\DeclareMathOperator{\SO}{SO}$$\DeclareMathOperator{\RP}{RP}$$\DeclareMathOperator{\Stab}{Stab}$There is a difference between the sets $C = [-1,1]^3$ and $E = [0,1]^3$.  Note that $\Stab(C)$, the stabilizer of $C$ inside of $\SO(3)$, is called the cube group and it has 24 elements.  Note that $\Stab(E)$ has three elements.


*

*Are you sure you want $E$ and not $C$?  


In either case, the manifold $X = X_E = \SO(3)/\Stab(E)$ is called a "spherical space form" and also an "elliptical manifold".  This is because $X$ is a quotient of the three-sphere by a freely acting subgroup of $\SO(4)$ -- note that $\SO(3)$ is a copy of $\RP^3$ so it is also a spherical space form.  Jeff Weeks' book "The shape of space" has an elementary treatment of the geometry and topology involved.  Also, here is link to Wikipedia:
https://en.wikipedia.org/wiki/Spherical_3-manifold
The manifold $X$ is called a "prism manifold" there.  Now, you ask

Does this space admit a simple parameterization, measure, or metric?

and you ask for 

[e.g. an embedding into Euclidean space without having to identify points] What does the induced Riemannian metric look like?

You haven't said why you are looking for this, beyond mentioning a computational application.  There are lots of embedding theorems (eg the Whitney embedding theorem) and there are ways to describe the induced metrics.  These will (almost certainly) be useless for computational purposes.
Probably you will be much happier with the following much simpler approach.   Encode elements of $X = \SO(3)/\Stab(E)$ as objects $[M]$ where 


*

*$M$ is an orthogonal matrix (of determinant one) and

*$[M] = [N]$ if and only if $N^{-1} M \in \Stab(E)$.


I'll guess that these new objects (basically, cosets) will do everything you want, while avoiding dangerous mucking about with wacky embeddings.  For example, computing distance in $X$ reduces to three distance computations in $\SO(3)$.
A: More useful than embedding it in some higher-dimensional space will be to give it as a manifold with identifications, I think. Topologically, $SO(3)/\!\sim$ is the same space as what you get by taking an octahedron and gluing the opposite faces with a $1/3$ turn. This is related to the 24-cell, one of the 6 different regular 4-dimensional polytopes (analogues of Platonic solids in 4 dimensions), using the fact that $SO(3)$ is topologically $RP^3$, the quotient of the 3-sphere $S^3$ by the antipodal map.
The space of course comes with a natural metric and measure (both inherited from the one on $SO(3)$. You can see this metric/measure on the octahedron by embedding it as a regular octahedron of an appropriate size subset of $SO(3)$. You can work out the size by looking at the volume: 12 of these octahedra tile $SO(3)$, or 24 tile the double-cover $S^3$. Alternatively, each corner of the octahedron in $SO(3)$ is a rotation by $\pi/4$ around one of the faces of the cube, and the center of each face of the octahedron is a rotation by $\pi/3$ around one of the vertices of the cube.
A: $SO(3)$ acts transitively on the space of all orthonormal oriented bases (these are ordered up to an even permutation). The stabilizer of the standard basis is exactly your subgroup $H$. So $X = SO(3)/\sim$ is the space of such bases. It inherits an $SO(3)$-invariant measure and Riemannian metric from $SO(3)$, which is actually a covering space of $X$ of degree $3$.
