The driven quantum asymmetric top is an important paradigm in molecular physics with applications ranging from quantum information to chiral-sensitive spectroscopy. A key prerequisite for these applications is the ability to completely control the rotational dynamics. The inherent degeneracy of quantum rotors poses a challenge for quantum control since selecting a particular rotational state cannot be achieved by spectral selection alone. Here, we prove complete controllability for rotational states of an asymmetric top belonging to degenerate values of the orientational quantum number M. Based on this insight, we construct a pulse sequence that energetically separates population in degenerate M-states. Introducing the concept of enantio-selective controllability, we determine the conditions for complete enantiomer-specific population transfer in chiral molecules and construct pulse sequences for the example of propanediol and carvone molecules for population initially distributed over degenerate M-states. Our work shows how to leverage controllability analysis for the solution of practical quantum control problems.