What Is Identified in Causal Representation Learning? Exact Answers for the Normalized iVAE from Realized Symmetries

Causal representation learning shows that a latent representation can be recovered only up to a symmetry. This leaves a question unanswered: which claims about hidden causes are fixed by observed data? We introduce a framework that answers it by characterizing functions of latent variables that are determined by observations and unchanged by symmetries. We first show why normalization is essential: a class that admits monotone changes of every latent coordinate identifies only constant functions. For the normalized iVAE with scalar statistics and a fixed latent parameterization, we build changes of variables that realize every range-preserving permutation and sign change. This turns the classical iVAE inclusion into an exact statement within each set of models producing the same observations: the bounded functions unaffected by those permutations and sign changes are identified. We show how these statements fit together across model classes. A rigid class preserves content, while a class allowing monotone changes in every coordinate reduces it to constants. Finally, we show how contrastive content and style separation fits the same interface and sketch how CITRIS fits the picture. Experiments with fitted iVAEs show that the symmetry is real in practice, that identified functions are the stable ones across retraining, and that a stronger normalization restores individual coordinates.