  We study cosmological perturbation spectra using the dynamical equations of
gauge invariant perturbations with a generalized blue/red-shift term. Combined
with the power-law index of cosmological background, {\nu}, we construct a
parameter space, ({\nu}, m), to classify all possible perturbation spectra. The
magnitude-invariant power spectra occupy a two dimensional region in the
({\nu}, m) space. We find two groups of scale-invariant solutions, one of which
are also magnitude invariant while the other not, in the expanding and
contracting phases of cosmological evolution. We explore the implications of
magnitude invariance to the scale-invariance of an underlying spectrum, and
unveil a novel duality between the scale-invariant solutions and the boundary
of magnitude-invariant solutions: under two consecutive duality
transformations, the scale-invariant solutions are mapped onto the boundary of
magnitude invariant region. The physical origin of such a duality is yet to be
studied. We present cosmological applications of our observations to a
de-Sitter universe, an Ekpyrotic background, and a matter-dominated
contraction. For the first two cases previously known results are re-derived;
we also find for a matter-dominated contraction a solution with both
scale-invariance as well as magnitude-invariance--in contrast to Wands's
scale-invariant but magnitude-variant spectrum.
