I know it is computationally a bit of a pain, but if you have points v_1,v_2,v_3 does the Jordan Canonical Form of the matrix [v_1 v_2 v_3] not satisfy your requirements? I believe that the original matrix is similar to both its permutations and rotations, so the JCF should be invariant. My linear algebra is a bit rusty, so I might be wrong