  A harmonic oscillator with time-dependent mass $m(t)$ and a time-dependent
(squared) frequency $\omega^2(t)$ occurs in the modelling of several physical
systems. It is generally believed that systems, with $m(t)>0$ and
$\omega^2(t)>0$ (normal oscillator) are stable while systems with $m(t)>0$ and
$\omega^2(t)<0$ (inverted oscillator) are unstable. We show that it is possible
to represent the \textit{same} physical system either as a normal oscillator or
as an inverted oscillator by redefinition of dynamical variables. While we
expect the physics to be invariant under such redefinitions, it is not obvious
how this invariance actually comes about. We study the relation between these
two, normal and inverted, representations of an oscillator in detail both in
Heisenberg and Schr\"{o}dinger pictures to clarify several conceptual and
technical issues. The situation becomes more involved when the oscillator is
coupled to another (semi)classical degree of freedom $C(t)$ and we want to
study the back-reaction of the quantum system $q(t)$ on $C(t)$, in the
semi-classical approximation. We provide a simple prescription for the
back-reaction based on energy conservation and study that the dynamics of the
full system in both normal and the inverted oscillator representation. The
physics again remains invariant but there are some extra subtleties which we
clarify. The implications of these results for quantum field theory in
cosmological backgrounds are discussed briefly in an appendix.
