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Natural and synthetic multicomponent gels display emergent properties, which implies that they are more than just the sum of their components. This warrants the investigation of the role played by interspecies interactions in shaping gel architecture and rheology. Here, using computer simulations, we investigate the effect of changing the strength of the interactions between two species forming a fibrous double network. Simply changing the strength of interspecies lateral association, we generate two types of gels: one in which the two components demix and another one in which the two species wrap around each other. We show that demixed gels have structure and rheology that are largely unaffected by the strength of attraction between the components. In contrast, architecture and material properties of intertwined gels strongly depend on interspecies “stickiness” and volume exclusion. These results can be used as the basis of a design principle for double networks which are made to emphasize either stability to perturbations or responsiveness to stimuli. Similar ideas could be used to interpret naturally occurring multicomponent gels.