Let me begin by saying that I am a mechanical engineering student without much of a math background. I am attempting to write a program which requires a selection of unique cube orientations sampled from all orientations as uniformly as possible... no repeated orientations are allowed and the selected orientations should not be too similar (perhaps something like a minimum $5^\circ$ separation about any axis). The cube is completely uniform, so I define uniqueness between two given orientations by whether or not a symmetry transform relates them. If they are related by a symmetry transform - they are not unique. I am aware that the cube symmetry group is quite large with 24 rotational symmetries existing for any given orientation.
In my mind, I picture the 3D rotation group as a vector space with elements $(\theta_x,\theta_y,\theta_z)$. In trying to tackle this problem, I am attempting to imagine what that vector space would look like with all the rotations that would lead to duplicate cube orientations deleted. If an asymmetric object's unique orientations completely fill the 3D rotation group space, what portions can be deleted in the case of a cubical object?
I am also aware that the 3D rotation group can be represented by the space of $3\times 3$ orthogonal matrices. Perhaps there are some ideas from linear algebra that may be directly applicable to this problem.
More Concisely (Thank you user7530)
For a uniform cube, define sufficiently unique cube orientations as those that have significant difference in orientation under all symmetry transformations. Does $SO(3)/O$, where $O$ is the octahedral group, have a known structure that may facilitate the selection of sufficiently unique cube orientations?
Any advice would be greatly appreciated!