  We provide a way of generating and observing molecular quantum gyroscopic
motion that resembles gyroscopic motion of classical rotors. After producing a
nonspreading rotational wavepacket 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 a method to measure rotational g factors of molecules.
