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We characterize how the response of physical systems to spatial rotations constrains the probabilities of events that may be observed. From a foundational point of view, we show that the set of quantum correlations in our scenarios can be derived from rotational symmetry alone, without assuming quantum physics. This shows that important predictions of quantum theory can be derived from the structure of space, demonstrating that semi-device-independent scenarios can be utilized to shed light on the foundations of physics. From a practical perspective, these results allow us to introduce semi-device-independent protocols for the generation of secure random numbers based on the breaking of spatial symmetries. While experimental implementations will rely on quantum physics, the security analysis and the amount of extracted randomness is theory-independent and certified by the observed correlations only. That is, our protocols rely on a physically meaningful assumption: a bound on a theory-independent notion of spin.