  Low-lying bandhead states in axially prolate deformed odd-odd nuclei have
long been described essentially within the rotor+two-quasiparticle picture.
This approach allows one to explain the appearance of so-called
Gallagher-Moszkowski doublets of bandheads with $K = \Omega_n \pm \Omega_p$,
sum and difference of neutron and proton angular momentum projections on the
symmetry axis. According to an empirical rule stated by Gallagher and
Moszkowski the spin-aligned configuration lies lower in energy than the
spin-anti-aligned configuration. A recent study by Robledo, Bernard and Bertsch
in Phys. Rev. C 89, 021303(R) (2014) within the Gogny energy-density functional
with selfconsistent blocking of the unpaired nucleons showed that calculations
fail to reproduce this rule in about half of the cases and points to the
density-dependent term of the functional as responsible of this failure.
  In this paper we aim at pushing further this analysis to exhibit the
mechanism underlying the energy splitting in a Gallagher-Moszkowski doublet. We
work in the framework of the Skyrme energy-density functional approach,
including BCS pairing correlations with selfconsistent blocking. We use the
SIII parametrization with time-odd terms and seniority pairing matrix elements
extending a previous study of K-isomeric states in even-even nuclei [Phys. Rev.
C 105, 044329 (2022)].
  We find that the energy splitting results from a competition between the
spin-spin, density-dependent and current-current terms of the Skyrme
energy-density functional. In doublets where the larger K value is lower in
energy the Gallagher-Moszkowski rule is always satisfied by the SIII Skyrme
energy-density functional. In doublets, on the contrary, where the smaller K
value lies lower, the energy splittings are calculated to be rather small and
often a disagreement with the Gallagher-Moszkowski rule occurs.
