  A switching random walk, commonly known under the misnomer `oscillating
random walk', is a real-valued Markov chain such that the distribution of its
increments depends only on the sign of the current position. In this note we
find invariant measures for such chains. In the particular case where the chain
is an actual random walk, our proof naturally relates its stationarity relative
to the Lebesgue measure to stationarity of the renewal processes of its
ascending and descending ladder heights.
