First, a "canonical" dice numbering, for this purpose, is one where opposite sides sum to 13.
We can interpret all possible numberings as permutations of pairs of opposing faces, as well as the permutations within those pairs. This gives $2^6\times6!=46080$ possible combinations (regardless of rotational symmetry), described by the group $C_2\wr S_6$.
There are 60 rotational symmetries of a dodecahedron, described by the group $I$ (or $A_5$). There is only one element of $I$ (the identity) that will preserve any of the permutations; all other elements will modify the permutations in some way. By Burnside's Lemma, that means there are $\frac{46080}{60}=768$ canonical dice numberings up to rotational symmetry.
This answer leaves me somewhat unsatisfied, however. It's good to have a number, but it isn't very intuitive to me. I'd really like some way to describe what these distinct combinations actually are, and that brings me to my question:
Is there a group, or a group-like object, that describes the canonical dice numberings of a dodecahedral dice up to rotational symmetry? Or, alternatively, is there a combinatorical method of constructing these dice numberings?
Sorry if this post is difficult to understand. I don't have a formal background at this level so my word choice might be a little off.
Best Answer
Fix the orientation of the $(1,12)$ pair so that the $1$ faces straight up, then rotate about the $z$-axis so that the $(2,11)$ pair is pointing upwards and away from you. Then there are $4!=24$ ways to arrange the remaining pairs $(3,10)(4,9)(5,8)(6,7)$ (their upper numbers form a line around the $1$) and $2^5=32$ ways to decide which way the smaller numbers of pairs $(2,11)$ to $(6,7)$ point, for a total of $768$ canonical numberings.