Relative symplectic cohomology, a functor from compact subsets of a symplectic manifold to modules, was recently introduced by Varolgunes. After passing to the torsion-free part, this invariant has been computed in various cases and has led to several new results in symplectic topology. However, due to the complexity of the construction, the torsion component has remained largely unexplored, and its nontriviality in general has been conjectural. In this talk, I will present several computations of relative symplectic cohomology that include the torsion part, prove that the torsion is indeed nontrivial, and use it to exhibit the first example of a new symplectic rigidity phenomenon. This talk is based on joint works in progress with Yaniv Ganor and Frol Zapolsky.