A reverse-flow reactor (RFR) usually attains a symmetric period-1 state, so that the temperature profile just after a flow reversal is a mirror image of that after the previous Flaw reversal. Recent simulations show that in certain cases a cooled RFR may attain other states with different types of periodicity or even complex quasi-periodic states. The maximum temperature of these states often exceeds that of the symmetric states so that it may deactivate the catalyst and/or lend to safety problems. A systematic, numerically efficient method is presented for constructing maps of parameter regions in which a cooled RFR has qualitatively different dynamic features. The technique is applied to determine the dependence of these dynamic features on the cooling capacity and flow-reversal period. Stable quasi-periodic and asymmetric period-1 states exist mainly for short flow-reversal periods. The quasi-periodic states usually exist for lower cooling capacities than those for which the asymmetric period-1 states exist. Stable symmetric and asymmetric period-1 states exist for the same set of parameters in very narrow regions of the parameter space. The behavior of the RFR in the limit of very fast flow reversals is usually modeled by a countercurrent flow reactor, with equal flow rates in the two compartments. This cooled reactor may attain asymmetric states for certain sets of parameters.