  We investigate the problem of the extraction of the isovector and isoscalar
spectral functions from data on $e^+e^-\to hadrons$, in the presence of
non-zero isospin breaking. It is shown that the conventional approach to
extracting the isovector spectral function in the $\rho$ resonance region, in
which only the isoscalar contribution associated with $\omega\to \pi\pi$ is
subtracted, fails to fully remove the effects of the isoscalar component of the
electromagnetic current. The additional subtractions required to extract the
pure isovector and isoscalar spectral functions are estimated using results
from QCD sum rules. It is shown that the corrections are small ($\sim 2%$) in
the isovector case (though relevant to precision tests of CVC), but very large
($\sim 20%$) in the case of the $\omega$ contribution to the isoscalar spectral
function. The reason such a large effect is natural in the isoscalar channel is
explained, and implications for other applications, such as the extraction of
the sixth order chiral low-energy constant, $Q$, are discussed.
