  A theoretical study is presented of the effect of an in-plane magnetic
exchange field on the band structure of centrosymmetric films of noble metals
and topological insulators. Based on an ab initio relativistic
$\mathbf{k}\cdot\mathbf{p}$ theory, a minimal effective model is developed that
describes two coupled copies of a Rashba or Dirac electronic system residing at
the opposite surfaces of the film. The coupling leads to a structural gap at
$\bar{\Gamma}$ and causes an exotic redistribution of the spin density in the
film when the exchange field is introduced. We apply the model to a
nineteen-layer Au(111) film and to a five-quintuple-layer Sb$_2$Te$_3$ film. We
demonstrate that at each film surface the exchange field induces spectrum
distortions similar to those known for Rashba or Dirac surface states with an
important difference due to the coupling: At some energies, one branch of the
state loses its counterpart with the oppositely directed group velocity. This
suggests that a large-angle electron scattering between the film surfaces
through the interior of the film is dominant or even the only possible for such
energies. The spin-density redistribution accompanying the loss of the
counterpart favors this scattering channel.
