Directed cycles form the fundamental motifs in natural, social and artificial networks, yet their distinct computational roles remain under-explored, particularly in the context of higher-order structure and function. In this work, we investigate how two types of directed cycles - feedforward and feedback - can act as higher-order structures to facilitate the flow and integration of information in sparse random networks, and how these roles depend on the environment of the cycles. Using information-theoretic measures, we show that network size, sparsity and relative directionality critically impact the information-processing capacities of directed cycles. In a network with no-preferred global direction, a feedforward cycle enables greater information flow and a feedback cycle allows for increased information integration. The relative direction of a feedforward cycle as well as the structural incoherence it induces, determines its capacity to generate higher-order behaviour. Finally, we demonstrate that introducing feedback loops into otherwise feedforward architectures increases the diversity of network activity patterns. These findings suggest that directed cycles serve as computational motifs with local information processing capabilities that depend on the structure they are embedded. Using directed cycles, we highlight the interdependence between higher-order structures and the higher-order order behaviour they can induce in the network dynamics.