We investigate relationships between interleaving and causal notions of game semantics for concurrent programming languages, focusing on the existence of canonical compact causal representations of the interleaving game semantics of programs. We perform our study on an affine variant of Idealized Parallel Algol (IPA), for which we present two games model: an interleaving model (an adaptation of Ghica and Murawski's fully abstract games model for IPA up to may-testing), and a causal model (a variant of Rideau and Winskel's games on event structures). Both models are sound and adequate for affine IPA. Then, we relate the two models. First we give a causality-forgetting operation mapping functorially the causal model to the interleaving one. We show that from an interleaving strategy we can reconstruct a causal strategy, from which it follows that the interleaving model is the observational quotient of the causal one. Then, we investigate several reconstructions of causal strategies from interleaving ones, showing finally that there are programs which are inherently causally ambiguous, with several distinct minimal causal representations.