We discuss a semiclassical treatment of the rigid asymmetric rotor that delivers eigenenergies as well as eigenstates. We give possibilities to improve the semiclassical wave functions to any accuracy required. The method is devised so that inclusion of vibrations is possible. As no information about energetically lower states is included in the procedure, the calculation of highly excited states is easier than with conventional quantum methods. Calculation of quantum splitting from semiclassical eigenstates is treated. We give numerical examples for every procedure developed, so that the performance of the theory can be judged.