  We introduce a class of semi-device-independent protocols based on the
breaking of spacetime symmetries. In particular, we characterise how the
response of physical systems to spatial rotations constrains the probabilities
of events that may be observed: in our setup, the set of quantum correlations
arises from rotational symmetry without assuming quantum physics. On a
practical level, our results allow for the generation of secure random numbers
without trusting the devices or assuming quantum theory. On a fundamental
level, we open a theory-agnostic framework for probing the interplay between
probabilities of events (as prevalent in quantum mechanics) and the properties
of spacetime (as prevalent in relativity).
