A time-reversible molecular dynamics algorithm is presented for rigid bodies in the quarternion representation. The algorithm is developed on the basis of the Trotter factorization scheme, and its structure is similar to that of the velocity Verlet algorithm. When the rigid body is an asymmetric top, its computationally inconvenient Eulerian equation of motion is integrated by combining the computationally convenient solutions to the Eulerian equations of motion for two symmetric tops. It is shown that a larger time step is allowed in the time-reversible algorithm than in the Gear predictor-corrector algorithm. The efficiency of the hybrid Monte Carlo method for a molecular system is also examined using the time-reversible molecular dynamics algorithm in the quarternion representation.