  We investigate the role of the Coulomb and the vector-isovector $\rho$
potentials in the asymmetry of the neutron and proton pseudospin splittings in
nuclei. To this end, we solve the Dirac equation for the nucleons using central
vector and scalar potentials with Woods-Saxon shape and $Z$ and $N-Z$ dependent
Coulomb and $\rho$ potentials added to the vector potential. We study the
effect of these potentials on the energy splittings of proton and neutron
pseudospin partners along a Sn isotopic chain. We use an energy decomposition
proposed in a previous work to assess the effect of a pseudospin-orbit
potential on those splittings. We conclude that the effect of the Coulomb
potential is quite small and the $\rho$ potential gives the main contribution
to the observed isospin asymmetry of the pseudospin splittings. This isospin
asymmetry results from a cancellation of the various energy terms and cannot be
attributed only to the pseudospin-orbit term, confirming the dynamical
character of this symmetry pointed out in previous works.
