We propose a kinetic model describing a polyatomic gas undergoing resonant collisions, in which the microscopic internal and kinetic energies are separately conserved during a collision process. This behaviour has been observed in some physical phenomena, for example in the collisions between selectively excited CO2 molecules. We discuss the model itself, prove the related H-theorem and show that, at the equilibrium, two temperatures are expected. We eventually present a numerical illustration of the model and its main properties.