Given two sets of points $A$ and $B$ in $\mathbb R^n$, is there an efficient way to tell if $B$ is just a rotation of $A$ about some fixed point?

Question

For two sets of points in $\mathbb R^n$, $A$ and $B$, is there an efficient way to determine if we can rotate $B$ onto $A$?

Edit: Thanks for all the comments!

I wanted to be as general as possible, but we can restrict $A$ and $B$ to be the extreme points of a convex $n$-polytope, changing the problem to polytope isomorphism. If you all think this is too big of a change in the spirit of the question, I'll make a new one

Also I do mean computable in polynomial time when I ask for efficiency. I'm optimistic that there's a solution that's based on |$A$| and is invariant to $n$