By using direct numerical simulations at unprecedented resolution we study turbulence under rotation in presence of simultaneous direct and inverse cascades. The accumulation of energy at large scale leads to the formation of vertical coherent regions with high vorticity oriented along the rotation axis. By seeding the flow with millions of inertial particles we quantify -for the first time- the effects of those coherent vertical structures on the preferential concentration of light and heavy particles. Furthermore, we quantitatively show that extreme fluctuations, leading to deviations from a normal-distributed statistics, result from the entangled interaction of the vertical structures with the turbulent background. Finally, we present the first --ever-- measurement of the relative importance between Stokes drag, Coriolis and centripetal forces along the trajectories of inertial particles. We discovered that vortical coherent structures lead to unexpected diffusion properties for heavy and light particles in the directions parallel and perpendicular to the rotation axis.