The interaction of intruding objects with deformable materials arises in many contexts, including locomotion in fluids and loose media, impact and penetration problems, and geospace applications. Despite the complex constitutive behaviour of granular media, forces on arbitrarily shaped granular intruders are observed to obey surprisingly simple, yet empirical 'resistive force hypotheses'. The physics of this macroscale reduction, and how it might play out in other media, has however remained elusive. Here, we show that all resistive force hypotheses in grains arise from local frictional yielding, revealing a novel invariance within a class of plasticity models. This mechanical foundation, supported by numerical and experimental validations, leads to a general analytical criterion to determine which rheologies can obey resistive force hypotheses. We use it to explain why viscous fluids are observed to perform worse than grains, and to predict a new family of resistive-force-obeying materials: cohesive media such as pastes, gels and muds.