  We study geodesic motion near the throats of asymptotically flat, static,
spherically~symmetric traversable wormholes supported by a self-gravitating
minimally coupled phantom scalar field with an arbitrary self-interaction
potential. We assume that any such wormhole possesses the reflection symmetry
with respect to the throat, and consider only its observable "right~half". It
turns out that the main features of bound orbits and photon trajectories close
to the throats of such wormholes are very different from those near the
horizons of black holes. We~distinguish between wormholes of two types, the
first and second ones, depending on whether the redshift metric function has a
minimum or maximum at the throat. First, it turns out that orbits located near
the centre of a wormhole of any type exhibit retrograde precession, that is,
the angle of pericentre precession is negative. Second, in the case of high
accretion activity, wormholes of the first type have the innermost stable
circular orbit at the throat while those of the second type have the
resting-state stable circular orbit in which test particles are at rest at all
times. In our study, we have in mind the possibility that the strongly
gravitating objects in the centres of galaxies are wormholes, which can be
regarded as an alternative to black holes, and the scalar field can be regarded
as a realistic model of dark matter surrounding galactic centres. In this
connection, we~discuss qualitatively some observational aspects of results
obtained in this article.
