  For a 4-dimensional spatially-flat Friedmann-Robertson-Walker universe with a
scalar field $\phi(x)$, potential $V(\phi)$ and constant equation of state
$w=p/\rho$, we show that an expanding solution characterized by
$\epsilon=3(1+w)/2$ produces the same scalar perturbations as a contracting
solution with $\hat{\epsilon}=1/\epsilon$. The same symmetry applies to both
the dominant and subdominant scalar perturbation modes. This result admits a
simple physical interpretation and generalizes to $d$ spacetime dimensions if
we define $\epsilon \equiv [(2d-5)+(d-1)w]/(d-2)$.
