  The Newton--Hooke duality and its generalization to arbitrary power laws in
classical, semiclassical and quantum mechanics are discussed. We pursue a view
that the power-law duality is a symmetry of the action under a set of duality
operations. The power dual symmetry is defined by invariance and reciprocity of
the action in the form of Hamilton's characteristic function. We find that the
power-law duality is basically a classical notion and breaks down at the level
of angular quantization. We propose an ad hoc procedure to preserve the dual
symmetry in quantum mechanics. The energy-coupling exchange maps required as
part of the duality operations that take one system to another lead to an
energy formula that relates the new energy to the old energy. The
transformation property of {the} Green function satisfying the radial
Schr\"odinger equation yields a formula that relates the new Green function to
the old one. The energy spectrum of the linear motion in a fractional power
potential is semiclassically evaluated. We find a way to show the
Coulomb--Hooke duality in the supersymmetric semiclassical action. We also
study the confinement potential problem with the help of the dual structure of
a two-term power potential.
