  We show that the ideas related to integrability and symmetry play an
important role not only in the string T-duality story but also in its point
particle counterpart. Applying those ideas, we find that the T-duality seems to
be a more widespread phenomenon in the context of the point particle dynamics
than it is in the string one; moreover, it concerns physically very relevant
point particle dynamical systems and not just somewhat exotic ones fabricated
for the purpose. As a source of T-duality examples, we consider maximally
superintegrable spherically symmetric electro-gravitational backgrounds in $n$
dimensions. We then describe in detail four such spherically symmetric
dynamical systems which are all mutually interconnected by a web of point
particle T-dualities. In particular, the dynamics of a charged particle
scattered by a repulsive Coulomb potential in a flat space is T-dual to the
dynamics of the Coulomb scattering in the space of constant negative curvature,
but it is also T-dual to the (conformal) Calogero-Moser inverse square dynamics
both in flat and hyperbolic spaces. Thus knowing just the Hamiltonian dynamics
of the scattered particle cannot give us an information about the curvature of
the space in which the particle moves.
