  Driven-dissipative quantum systems generically do not satisfy simple notions
of detailed balance based on the time symmetry of correlation functions. We
show that such systems can nonetheless exhibit a hidden time-reversal symmetry
which most directly manifests itself in a doubled version of the original
system prepared in an appropriate entangled thermofield double state. This
hidden time-reversal symmetry has a direct operational utility: it provides a
general method for finding exact solutions of non-trivial steady states.
Special cases of this approach include the coherent quantum absorber and
complex-$P$ function methods from quantum optics. We also show that hidden TRS
has observable consequences even in single-system experiments, and can be
broken by the non-trivial combination of nonlinearity, thermal fluctuations,
and driving. To illustrate our ideas, we analyze concrete examples of driven
qubits and nonlinear cavities. These systems exhibit hidden time-reversal
symmetry but not conventional detailed balance.
