The paper is concerned with a new approach for the recurrence property of the oscillating process on Z in Kemperman's sense. In the case when the random walk is ascending on Z − and descending on Z + , we determine the invariant measure of the embedded process of successive crossing times and then prove a necessary and sufficient condition for recurrence. Finally, we make use of this result to show that the general oscillating process is recurrent under some Hölder-typed moment assumptions.