We discuss the time-reversal behavior of dynamic cross-couplings among various hydrodynamic degrees of freedom in liquid crystal systems. Using a standard hydrodynamic description including linear irreversible thermodynamics, we show that the distinct thermodynamic requirements for reversible and irreversible couplings lead to experimentally accessible differences. We critically compare our descriptions with those of existing standard continuum mechanics theories, where time-reversal symmetry is not adequately invoked. The motivation comes from recent experimental progress allowing to discriminate between the hydrodynamic description and the continuum mechanics approach. This concerns the dynamics of Lehmann-type effects in chiral liquid crystals and the dynamic magneto-electric response in ferronematics and ferromagnetic nematics, a liquid multiferroic system. In addition, we discuss the consequences of time-reversal symmetry for flow alignment of the director in nematics (or pretransitional nematic domains) and for the dynamic thermo-mechanical and electro-mechanical couplings in textured nematic liquid crystals.