In this paper, we classify up to Hamiltonian isotopy Lagrangian tori that split as a product of circles in \(S^2 \times S^2\) , when the latter is equipped with a non-monotone split symplectic form. We show that this classification is equivalent to a problem of mathematical billiards in rectangles. We give many applications, among others: (1) answering a question on Lagrangian packing numbers raised by Polterovich–Shelukhin, (2) studying the topology of the space of Lagrangian tori, and (3) determining which split tori are images under symplectic ball embeddings of Chekanov or product tori in \(\mathbb {R}^4\) .
