  We study neutral Dirac particles confined to a family of $(1+2)$-dimensional
wormholes arising from surfaces of revolution with a constant negative Gaussian
curvature, in the framework of a comprehensive effective field theory allowing
deviations from Lorentz symmetry: the gravitational standard-model extension
(SME). The Dirac particles are described with a fixed background tensor field
that rules the Lorentz symmetry violation in the CPT-even gauge sector of SME.
We implement this geometrical approach by incorporating non-minimal couplings
that possibly induce a Lorentz-symmetry violating term in the modified Dirac
equation. We also analyze the exact analytical solutions of the corresponding
modified Dirac equation in the presence of a peculiar external magnetic field.
