We study random dynamical systems on the real line, considering each dynamical system together with the one generated by the inverse maps. We show that there is a duality between forward and inverse behaviour for such systems, splitting them into four classes (in terms of both dynamical and stationary measure aspects). This is analogous to the results already known for the smooth dynamics on [0,1], established in terms of the Lyapunov exponents at the endpoints; however, our arguments are purely topological, and thus our result is applicable to the general case of homeomorphisms of the real line.