Aside from rotation, there is another invariant between the two sets which is much simpler to calculate and allows it to become a matching problem. I think it's worth pointing this out because it is really possible on this kata to get lost in linear algebra.

I'm able to derive rotation matrix R between any pair o and c that are given in the answer, however this R is different for each answer pair! I have no idea what I'm doing wrong or how that's even possible.

I also feel this could benefit from some links describing the theory of the geometry. Although it's probably pretty basic from a 3d graphics point of view, I have no experience with that.