Rutherford scattering is usually described by treating the projectile either classically or as quantum mechanical plane waves. We treat them as localised wave packets and study their head-on collisions with the stationary target nuclei. A comprehensive study of the quantum solutions reveals that the paradigmatic Rutherford scattering experiment is asymmetric in time. This nonclassical feature is traced back to the convexity properties of Coulomb interaction. Finally, we sketch how these theoretical findings could be tested in experiments looking for the onset of nuclear reactions.