It was recently observed that Elo ratings fail at preserving transitive relations among strategies and therefore cannot correctly extract the transitive component of a game. We provide a characterization of transitive games as a weak variant of ordinal potential games and show that Elo ratings actually do preserve transitivity when computed in the right space, using suitable invertible mappings. Leveraging this insight, we introduce a new game decomposition of an arbitrary game into transitive and cyclic components that is learnt using a neural network-based architecture and that prioritises capturing the sign pattern of the game, namely transitive and cyclic relations among strategies. We link our approach to the known concept of sign-rank, and evaluate our methodology using both toy examples and empirical data from real-world games.