We show that a family of key periodic orbits drives the recollision process in a strong circularly polarized laser field. These orbits, coined recolliding periodic orbits, exist for a wide range of parameters, and their relative influence changes as the laser and atomic parameters are varied. We find the necessary conditions for recollision-driven nonsequential double ionization to occur. The outlined mechanism is universal in that it applies equally well beyond atoms: The internal structure of the target species plays a minor role in the recollision process.