  This paper combines two classical theories, namely metric projective
differential geometry and superintegrability. We study superintegrable systems
on 2-dimensional geometries that share the same geodesics, viewed as
unparametrized curves. We give a definition of projective equivalence of such
systems, which may be considered the projective analog of (conformal) St\"ackel
equivalence (coupling constant metamorphosis). Then, we discuss the
transformation behavior for projectively equivalent superintegrable systems and
find that the potential on a projectively equivalent geometry can be
reconstructed from a characteristic vector field. Moreover, potentials of
projectively equivalent Hamiltonians follow a linear superimposition rule. The
techniques are applied to several examples. In particular, we use them to
classify, up to St\"ackel equivalence, the superintegrable systems on
geometries with one, non-trivial projective symmetry.
