This paper presents a systematic framework for constructing generic models of nonequilibrium quantum dynamics with a target stationary (mixed) state. The framework identifies (almost) all combinations of Hamiltonian and dissipative dynamics that relax to a steady state of interest, generalizing the Davies' generator for dissipative relaxation at finite temperature to nonequilibrium dynamics targeting arbitrary stationary states. The authors focus on Gibbs states of stabilizer Hamiltonians, identifying local Lindbladians compatible therewith by constraining the rates of dissipative and unitary processes. Moreover, given terms in the Lindbladian not compatible with the target state, the formalism identifies the operations – including syndrome measurements and local feedback – one must apply to correct these errors. The methods also reveal new models of quantum dynamics, such as a "measurement-induced phase transition" where measurable two-point functions exhibit critical (power-law) scaling with distance at a critical ratio of the transverse field and rate of measurement and feedback. Time-reversal symmetry, defined naturally within the formalism, can be broken both in effectively classical and intrinsically quantum ways. The framework provides a systematic starting point for exploring the landscape of dynamical universality classes in open quantum systems, as well as identifying new protocols for quantum error correction. The paper also discusses the implications of time-reversal symmetry in both classical and quantum systems, and how it can be generalized to include additional transformations. The authors show how to construct both T-even and T-odd nontrivial dynamics that protect certain stabilizer states and how to correct different kinds of errors. The framework is applied to various quantum systems, including quantum error correction and the repetition code, and it is shown that the formalism can be used to generate both T-even and T-odd dynamics, with Davies' generator as a special case. The paper concludes with a discussion of the implications of the framework for understanding quantum systems and the potential applications in quantum computing and quantum information science.This paper presents a systematic framework for constructing generic models of nonequilibrium quantum dynamics with a target stationary (mixed) state. The framework identifies (almost) all combinations of Hamiltonian and dissipative dynamics that relax to a steady state of interest, generalizing the Davies' generator for dissipative relaxation at finite temperature to nonequilibrium dynamics targeting arbitrary stationary states. The authors focus on Gibbs states of stabilizer Hamiltonians, identifying local Lindbladians compatible therewith by constraining the rates of dissipative and unitary processes. Moreover, given terms in the Lindbladian not compatible with the target state, the formalism identifies the operations – including syndrome measurements and local feedback – one must apply to correct these errors. The methods also reveal new models of quantum dynamics, such as a "measurement-induced phase transition" where measurable two-point functions exhibit critical (power-law) scaling with distance at a critical ratio of the transverse field and rate of measurement and feedback. Time-reversal symmetry, defined naturally within the formalism, can be broken both in effectively classical and intrinsically quantum ways. The framework provides a systematic starting point for exploring the landscape of dynamical universality classes in open quantum systems, as well as identifying new protocols for quantum error correction. The paper also discusses the implications of time-reversal symmetry in both classical and quantum systems, and how it can be generalized to include additional transformations. The authors show how to construct both T-even and T-odd nontrivial dynamics that protect certain stabilizer states and how to correct different kinds of errors. The framework is applied to various quantum systems, including quantum error correction and the repetition code, and it is shown that the formalism can be used to generate both T-even and T-odd dynamics, with Davies' generator as a special case. The paper concludes with a discussion of the implications of the framework for understanding quantum systems and the potential applications in quantum computing and quantum information science.