  In this paper, we consider a stochastic process that may experience random
reset events which relocate the system to its starting position. We focus our
attention on a one-dimensional, monotonic continuous-time random walk with a
constant drift: the process moves in a fixed direction between the reset
events, either by the effect of the random jumps, or by the action of a
deterministic bias. However, the orientation of its motion is randomly
determined after each restart. As a result of these alternating dynamics,
interesting properties do emerge. General formulas for the propagator as well
as for two extreme statistics, the survival probability and the mean
first-passage time, are also derived. The rigor of these analytical results is
verified by numerical estimations, for particular but illuminating examples.
