  We study a planar random motion $\big(X(t),\,Y(t)\big)$ with orthogonal
directions, where the direction switches are governed by a homogeneous Poisson
process. At each Poisson event, the moving particle turns clockwise or
counterclockwise according to a rule which depends on the current direction. We
prove that the components of the vector $\big(X(t),\,Y(t)\big)$ can be
represented as linear combinations of two independent telegraph processes with
different intensities. The exact distribution of $\big(X(t),\,Y(t)\big)$ is
then obtained both in the interior of the support and on its boundary, where a
singular component is present. We show that, in the hydrodynamic limit, the
process behaves as a planar Brownian motion with correlated components. The
distribution of the time spent by the process moving vertically is then
studied. We obtain its exact distribution and discuss its hydrodynamic limit.
In particular, in the limiting case, the process $\big(X(t),\,Y(t)\big)$ spends
half of the time moving vertically.
