  Given an autohomeomorphism on an ordered topological space or its subspace,
we show that it is sometimes possible to introduce a new topology-compatible
order on that space so that the same map is monotonic with respect to the new
ordering. We note that the existence of such a re-ordering for a given map is
equivalent to the map being conjugate (topologically equivalent) to a monotonic
map on some homeomorphic ordered space. We observe that the latter cannot
always be chosen to be order-isomorphic to the original space. Also, we
identify other routes that may lead to similar affirmative statements for other
classes of spaces and maps.
