Stable operation of a reverse-flow reactor is accomplished as the so-called cyclic steady-state which is characterised by strictly defined symmetry properties of the profiles of state variables. These symmetry properties have been determined as a result of numerous dynamic simulations of the process considered. In the present study, introducing the symmetry properties into the balance equations of the mathematical model of the reactor, exact quantitative relations have been developed which clearly define the cyclic steady-state and, simultaneously, determine the criterion of autothermicity of the process. In the considerations, a one-dimensional, two-phase, continuous model of the reactor has been applied. The equations of the reactor model have been transformed to such a form which enables the introduction of the symmetry properties of the profiles of temperature and mixture composition. Moreover, the equations derived form a tool, which makes it possible to investigate, in a relatively simple way, the convergence to the cyclic steady-state during the dynamic simulations.