We study the evolution of two mutually interacting pairwise games on different topologies. On two-dimensional square lattices, we reveal that the game-game interaction can promote the cooperation prevalence in all cases, and the cooperation-defection phase transitions even become absent and fairly high cooperation is expected when the interaction becomes very strong. A mean-field theory is developed that points out dynamical routes arising therein. Detailed analysis shows indeed that there are rich categories of interactions in either the individual or bulk scenario: invasion, neutral, and catalyzed types; their combination puts cooperators at a persistent advantage position, which boosts the cooperation. The robustness of the revealed reciprocity is strengthened by the studies of model variants, including the public goods game, asymmetrical or time-varying interactions, games of different types, games with timescale separation, different updating rules, etc. The structural complexities of the underlying population, such as Newman-Watts small world networks, Erdős-Rényi random networks, and Barabási-Albert networks, also do not alter the working of the dynamical reciprocity. In particular, as the number of games engaged increases, the cooperation level continuously improves in general. However, our analysis shows that the dynamical reciprocity works only in structured populations, otherwise the game-game interaction has no any impact on the cooperation at all. In brief, our work uncovers a cooperation mechanism in the structured populations, which indicates the great potential for human cooperation since concurrent issues are so often seen in the real world.