A major challenge in systems biology is to understand the relationship between a circuit's structure and its function, but how is this relationship affected if the circuit must perform multiple distinct functions within the same organism? In particular, to what extent do multi-functional circuits contain modules which reflect the different functions? Here, we computationally survey a range of bi-functional circuits which show no simple structural modularity: They can switch between two qualitatively distinct functions, while both functions depend on all genes of the circuit. Our analysis reveals two distinct classes: circuits which overlay two simpler mono-functional sub-circuits within their circuitry, and circuits, which do not. In this second class, the bi-functionality emerges from more complex designs which are not fully decomposable into distinct modules and are consequently less intuitive to predict or understand. These non-intuitive emergent circuits are just as robust as their hybrid counterparts, and we therefore suggest that the common bias toward studying modular systems may hinder our understanding of real biological circuits.