I saw on wikipedia that the determinant of a rotation matrix is always one (possibly by definition?), but it doesn't say anything about the determinant of the Jacobian of such a matrix.
Since applying a rotation shouldn't change the integral of a rotationally invariant function over $\mathbb{R}^d$, if $(v_1,...,v_d) = \phi(u_1,...,u_d)$ is our rotation, then it should be the case that $d_{v_1}...d_{v_d} = |det(D \phi)(u)| d_{u_1}...d_{u_d} = d_{u_1}...d_{u_d}$ right (here $D$ stands for the Jacobian)?
I don't know enough about rotation matrices to make that justify that claim though.