  The $Kepler$ $orbits$ form a 3-parameter family of $unparametrized$ plane
curves, consisting of all conics sharing a focus at a fixed point. We study the
geometry and symmetry properties of this family, as well as natural 2-parameter
subfamilies, such as those of fixed energy or angular momentum.
  Our main result is that Kepler orbits is a `flat' family, that is, the local
diffeomorphisms of the plane preserving this family form a 7-dimensional local
group, the maximum dimension possible for the symmetry group of a 3-parameter
family of plane curves. These symmetries are different from the well-studied
`hidden' symmetries of the Kepler problem, acting on energy levels in the
4-dimensional phase space of the Kepler system.
  Each 2-parameter subfamily of Kepler orbits with fixed non-zero energy
(Kepler ellipses or hyperbolas with fixed length of major axis) admits
$\mathrm{PSL}_2(\mathbb{R})$ as its (local) symmetry group, corresponding to
one of the items of a classification due to A. Tresse (1896) of 2-parameter
families of plane curves admitting a 3-dimensional local group of symmetries.
The 2-parameter subfamilies with zero energy (Kepler parabolas) or fixed
non-zero angular momentum are flat (locally diffeomorphic to the family of
straight lines).
  These results can be proved using techniques developed in the 19th century by
S. Lie to determine `infinitesimal point symmetries' of ODEs, but our proofs
are much simpler, using a projective geometric model for the Kepler orbits
(plane sections of a cone in projective 3-space). In this projective model all
symmetry groups act globally.
  Another advantage of the projective model is a duality between Kepler's plane
and Minkowski's 3-space parametrizing the space of Kepler orbits. We use this
duality to deduce several results on the Kepler system, old and new.
