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Disentanglement via mechanism sparsity was introduced recently as a principled approach to extract latent factors without supervision when the causal graph relating them in time is sparse, and/or when actions are observed and affect them sparsely. However, this theory applies only to ground-truth graphs satisfying a specific criterion. In this work, we introduce a generalization of this theory which applies to any ground-truth graph and specifies qualitatively how disentangled the learned representation is expected to be, via a new equivalence relation over models we call consistency. This equivalence captures which factors are expected to remain entangled and which are not based on the specific form of the ground-truth graph. We call this weaker form of identifiability partial disentanglement. The graphical criterion that allows complete disentanglement, proposed in an earlier work, can be derived as a special case of our theory. Finally, we enforce graph sparsity with constrained optimization and illustrate our theory and algorithm in simulations.