Here, we develop a symmetry-“roadmap” for identifying photonic Chern phases in honeycomb crystals, via tight-binding theory. To this end, we study how a generic nonreciprocal perturbation of photonic graphene affects the system topology. It is shown that a topological gap can be opened up only when the nonreciprocal perturbation exhibits even parity under re- flections about the horizontal and vertical mirror axes of the hexagonal unit cell. To illustrate our theory, we analyze the topology of photonic graphene crystals modified with rings of gyro- magnetic inclusions, confirming in this way the fundamental role of symmetry in the emergence of topological phases.