  This paper studies a family of random walks defined on the finite ordinals
using their order reversing involutions. Starting at $x \in
\{0,1,\ldots,n-1\}$, an element $y \le x$ is chosen according to a prescribed
probability distribution, and the walk then steps to $n-1-y$. We show that
under very mild assumptions these walks are irreducible, recurrent and ergodic.
We then find the invariant distributions, eigenvalues and eigenvectors of a
distinguished subfamily of walks whose transition matrices have the global
anti-diagonal eigenvalue property studied in earlier work by Ochiai, Sasada,
Shirai and Tsuboi. We prove that this subfamily of walks is characterised by
their reversibility. As a corollary, we obtain the invariant distributions and
rate of convergence of the random walk on the set of subsets of $\{1,\ldots,
m\}$ in which steps are taken alternately to subsets and supersets, each chosen
equiprobably. We then consider analogously defined random walks on the real
interval $[0,1]$ and use techniques from the theory of self adjoint compact
operators on Hilbert spaces to prove analogues of the main results in the
discrete case.
