  The Douglas-Rachford algorithm is a popular method for finding zeros of sums
of monotone operators. By its definition, the Douglas-Rachford operator is not
symmetric with respect to the order of the two operators. In this paper we
provide a systematic study of the two possible Douglas-Rachford operators. We
show that the reflectors of the underlying operators act as bijections between
the fixed points sets of the two Douglas-Rachford operators. Some elegant
formulae arise under additional assumptions. Various examples illustrate our
results.
