Identifiable representation learning asks when a learned latent representation corresponds to generative, causal, or task-relevant factors rather than an arbitrary nonlinear reparameterization of the same observed distribution. This article develops anchor--stabilizer theory as a symmetry-breaking framework for this problem. The central idea is that positive identifiability results rely, explicitly or implicitly, on generalized anchors: conditional, temporal, structural, mechanistic, relational, semantic, objective-induced, decoder-level, or query-level resources that restrict the latent transformations left unresolved by the observed marginal distribution. An anchor is formalized as an augmented observation or constraint operator that refines baseline observational equivalence. The residual ambiguity left by the anchor is its stabilizer, namely the set of latent reparameterizations that preserve both the original observations and the anchored structure. Identifiability is obtained when this stabilizer is contained in an accepted ambiguity class, such as permutation, component-wise transformations, block transformations, graph-preserving transformations, or query-preserving transformations. Composite anchors reduce ambiguity by intersecting stabilizers, explaining why individually insufficient resources may become jointly identifying. The framework unifies auxiliary-variable nonlinear ICA, identifiable VAEs, structured nonlinear ICA, sparse and geometric decoders, causal representation learning, grouping, contrastive learning, augmentation invariance, supervised labels, and no-auxiliary approaches as instances of the same symmetry-breaking principle.