Systems whose organization displays causal asymmetry constraints, from evolutionary trees to river basins or transport networks, can often be described in terms of directed paths on a discrete set of arbitrary units including states in state spaces, feed-forward neural nets, the evolutionary history of a given collection of events or the chart of computational states visited along a complex computation. Such a set of paths defines a feed-forward, acyclic network. A key problem associated with these systems involves characterizing their intrinsic degree of path reversibility: given an end node in the graph, what is the uncertainty of recovering the process backwards until the origin? Here, we propose a novel concept, topological reversibility, which is a measure of the complexity of the net that rigorously weights such uncertainty in path dependency, quantifying the minimum amount of information required to successfully reverse a causal path. Within the proposed framework, we also analytically characterize limit cases for both topologically reversible and maximally entropic structures. The relevance of these measures within the context of evolutionary dynamics is highlighted.