  In this paper we consider the one-dimensional dynamical evolution of a
particle traveling at constant speed and performing, at a given rate, random
reversals of the velocity direction. The particle is subject to stochastic
resetting, meaning that at random times it is forced to return to the starting
point. Here we consider a return mechanism governed by a deterministic law of
motion, so that the time cost required to return is correlated to the position
occupied at the time of the reset. We show that in such conditions the process
reaches a stationary state which, for some kinds of deterministic return
dynamics, is independent of the return phase. Furthermore, we investigate the
first-passage properties of the system and provide explicit formulas for the
mean first-hitting time. Our findings are supported by numerical simulations.
