We provide away of generating and observing molecular quantum gyroscopic motion that resembles gyroscopic motion of classical rotors. After producing a nonspreading rotational wave packet called a cogwheel state, one can generate a gyroscopic precession motion by applying an external magnetic field interacting through a rotational magnetic dipole moment. The quantum rotors, realized with linear nonparamagnetic ionic molecules trapped in an ion trap, can keep their gyroscopic motion for a long time in a collectively synchronized fashion. A Coulomb-explosion technique is suggested to observe the gyroscopic motion. Despite limited molecular species, the observation of the gyroscopic motion can be adopted as amethod to measure rotational g factors of molecules. DOI: 10.1103/PhysRevA.87.032520