This paper investigates the classification of mixed-state topological orders in two spatial dimensions. The authors argue that mixed-state topological orders can be classified by their 1-form symmetries and associated anyon theories, which lead to a partial classification under two-way connectivity by quasi-local quantum channels. They provide examples based on topological subsystem codes, decohering G-graded string-net models, and "classically gauging" symmetry-enriched topological orders. One example is an Ising string-net model under dephasing noise. The authors study the resulting space of locally-indistinguishable states and compute modular transformations within a particular coherent space. They identify two possible effects of quasi-local quantum channels on anyon theories: (1) anyons can be incoherently proliferated, reducing to a commutant of the proliferated anyons, or (2) the system can be "classically gauged," resulting in the symmetrization of anyons and an extension by transparent bosons. Based on these mechanisms, the authors conjecture that mixed-state topological orders are classified by premodular anyon theories, i.e., those for which the braiding relations may be degenerate. The paper also discusses the equivalence relation on mixed states, locally indistinguishable states, and the classification of mixed-state topological orders in terms of premodular anyon theories.This paper investigates the classification of mixed-state topological orders in two spatial dimensions. The authors argue that mixed-state topological orders can be classified by their 1-form symmetries and associated anyon theories, which lead to a partial classification under two-way connectivity by quasi-local quantum channels. They provide examples based on topological subsystem codes, decohering G-graded string-net models, and "classically gauging" symmetry-enriched topological orders. One example is an Ising string-net model under dephasing noise. The authors study the resulting space of locally-indistinguishable states and compute modular transformations within a particular coherent space. They identify two possible effects of quasi-local quantum channels on anyon theories: (1) anyons can be incoherently proliferated, reducing to a commutant of the proliferated anyons, or (2) the system can be "classically gauged," resulting in the symmetrization of anyons and an extension by transparent bosons. Based on these mechanisms, the authors conjecture that mixed-state topological orders are classified by premodular anyon theories, i.e., those for which the braiding relations may be degenerate. The paper also discusses the equivalence relation on mixed states, locally indistinguishable states, and the classification of mixed-state topological orders in terms of premodular anyon theories.