A split torus in S² × S² is a Lagrangian torus obtained as the product of circles in the factors. The goal of this talk is to give a classification up to symplectomorphisms of such tori and illustrate that interesting things happen in case the symplectic form is non-monotone. Among other applications, we will answer a question about Lagrangian packings posed by Polterovich-Shelukhin. This is partially based on joint work with Joontae Kim.