In 1996, Bertoin and Werner [5] demonstrated a functional limit theorem, characterising the windings of pla- nar isotropic stable processes around the origin for large times, thereby complementing known results for planar Brownian mo- tion. The question of windings at small times can be handled us- ing scaling. Nonetheless we examine the case of windings at the the origin using new techniques from the theory of self-similar Markov processes. This allows us to understand upcrossings of (not necessarily symmetric) stable processes over the origin for large and small times in the one-dimensional setting.