One of the first exoplanet hosts, discovered about thirty years ago, the star 55 Cnc has been continuously observed ever since. It is now known to host at least five planets with orbital periods ranging from 17 hours to 15 years. It is also one of the most extreme metal-rich stars in the neighbourhood, and it has a low-mass secondary star. In this article, we present data obtained at the Canada-France-Hawaii Telescope with the SPIRou spectropolarimeter on both components of the 55 Cnc stellar system. We revisit the long-period radial-velocity signals of 55 Cnc A, with a focus on the role of the magnetic cycle, and propose the existence of a sixth planet candidate, whose period falls close to that of the magnetic cycle, or half of it. The other massive outer planet has a revised period of 13.15 years and a minimum mass of 3.8M_Jup_. Although some uncertainty remains about these outer planets, the characterisation of the four inner planets is very robust through the combination of many different datasets, and all signals are consistent in the near-infrared (nIR) and optical domains. In addition, the magnetic topology of the solar-type primary component of the system was observed by SPIRou at the minimum of its activity cycle, characterised by an amplitude ten times smaller than observed that during its maximum in 2017. For the low-mass component 55 Cnc B, we report the discovery of two exoplanets in the system, with a period of 6.799+/-0.0014 and 33.75+/-0.04 days and a minimum mass of 3.5+/-0.8 and 5.3+/-1.4M_{Earth}_, respectively. The secondary magnetic field is very weak, and the current dataset does not allow its precise characterisation, setting an upper limit of 10 G. The system 55 Cnc stands out as the sixth binary system with planetary systems around both components and the first one with non-equal-mass stellar components.