  The total anisotropy of a diffuse background composed of two or more sources,
such as the Fermi-LAT--measured gamma-ray background, is set by the anisotropy
of each source population and the contribution of each population to the total
intensity. The total anisotropy as a function of energy (the anisotropy energy
spectrum) will modulate as the relative contributions of the sources change,
implying that the anisotropy energy spectrum also encodes the intensity
spectrum of each source class. We develop techniques, applicable to any such
diffuse background, for unraveling the intensity spectrum of each component
source population given a measurement of the total intensity spectrum and the
total anisotropy energy spectrum, without introducing \emph{a priori}
assumptions about the spectra of the source classes. We demonstrate the
potential of these methods by applying them to example scenarios for the
composition of the Fermi-LAT gamma-ray background consistent with current data
and feasible within 10 years of observation.
