  Directed topology was introduced as a model of concurrent programs, where the
flow of time is described by distinguishing certain paths in the topological
space representing such a program. Algebraic invariants which respect this
directedness have been introduced to classify directed spaces. In this work we
study the properties of such invariants with respect to the reversal of the
flow of time in directed spaces. Known invariants, natural homotopy and
homology, have been shown to be unchanged under this time-reversal. We show
that these can be equipped with additional algebraic structure witnessing this
reversal. Specifically, when applied to a directed space and to its reversal,
we show that these refined invariants yield dual objects. We further refine
natural homotopy by introducing a notion of relative directed homotopy and
showing the existence of a long exact sequence of natural homotopy systems.
