  Timelike geodesics on a hyperplane orthogonal to the symmetry axis of the
G\"odel spacetime appear to be elliptic-like if standard coordinates naturally
adapted to the cylindrical symmetry are used. The orbit can then be suitably
described through an eccentricity-semi-latus rectum parametrization, familiar
from the Newtonian dynamics of a two-body system. However, changing coordinates
such planar geodesics all become explicitly circular, as exhibited by Kundt's
form of the G\"odel metric. We derive here a one-to-one correspondence between
the constants of the motion along these geodesics as well as between the
parameter spaces of elliptic-like versus circular geodesics. We also show how
to connect the two equivalent descriptions of particle motion by introducing a
pair of complex coordinates in the 2-planes orthogonal to the symmetry axis,
which brings the metric into a form which is invariant under M\"obius
transformations preserving the symmetries of the orbit, i.e., taking circles to
circles.
