  We study symmetries between untwisted and twisted strings on asymmetric
orbifolds. We present a list of asymmetric orbifold models to possess
intertwining currents which convert untwisted string states to twisted ones,
and vice versa. We also present a list of heterotic strings on asymmetric
orbifolds with supersymmetry between untwisted and twisted string states. Some
of properties inherent in asymmetric orbifolds, which are not shared by
symmetric orbifolds, are pointed out.
