Reversible time travel with freedom of choice

General relativity allows for the existence of closed time-like curves, along which a material object could travel back in time and interact with its past self. This possibility raises the question whether certain initial conditions, or more generally local operations, lead to inconsistencies and should thus be forbidden. Here we consider the most general deterministic dynamics connecting classical degrees of freedom defined on a set of bounded space-time regions, requiring that it is compatible with arbitrary operations performed in the local regions. We find that any such dynamics can be realised through reversible interactions. We further find that consistency with local operations is compatible with non-trivial time travel: three parties can interact in such a way to be all both in the future and in the past of each other, while being free to perform arbitrary local operations.