  We formulate N-fold supersymmetry in quantum mechanical systems with
reflection operators. As in the cases of other systems, they possess the two
significant characters of N-fold supersymmetry, namely, almost isospectrality
and weak quasi-solvability. We construct explicitly the most general 1- and
2-fold supersymmetric quantum mechanical systems with reflections. In the case
of N=2, we find that there are seven inequivalent such systems, three of which
are characterized by three arbitrary functions having definite parity while the
other four of which are by two. In addition, four of the seven inequivalent
systems do not reduce to ordinary quantum systems without reflections.
Furthermore, in certain particular cases, they are essentially equivalent to
the most general two-by-two Hermitian matrix 2-fold supersymmetric quantum
systems obtained previously by us.
