We employ a reverse-engineering approach to illuminate the neurocomputational building blocks that combine to support controlled semantic cognition: the storage and context-appropriate use of conceptual knowledge. By systematically varying the structure of a computational model and assessing the functional consequences, we identified the architectural properties that best promote some core functions of the semantic system. Semantic cognition presents a challenging test case, as the brain must achieve two seemingly contradictory functions: abstracting context-invariant conceptual representations across time and modalities, while producing specific context-sensitive behaviours appropriate for the immediate task. These functions were best achieved in models possessing a single, deep multimodal hub with sparse connections from modality-specific regions, and control systems acting on peripheral rather than deep network layers. The reverse-engineered model provides a unifying account of core findings in the cognitive neuroscience of controlled semantic cognition, including evidence from anatomy, neuropsychology and functional brain imaging.