Periodic, quasiperiodic and chaotic symmetric and unsymmetric spatiotemporal temperature patterns arise in the heterogeneous (two-phase) model of the catalytic reactor with periodic flow reversal. Coexistence of various types of stable solutions was observed. Chaotic solutions arising via the torus breaking route were also observed. The transitions among individual types of spatiotemporal patterns and their development are studied by means of Poincare maps of the associated model represented as a one-parameter family of discrete dynamical systems derived from the simulation results of the original continuous model.