  We use the BLM scale-fixing prescription to derive a renormalization-scheme
invariant relation between the coefficient function for the Bjorken sum rule
for polarized deep inelastic scattering and the $R$-ratio for the $e^+e^-$
annihilation cross section. This relation provides a generalization of the
Crewther relation to non-conformally invariant gauge theories. The derived
relations allow one to calculate unambiguously without renormalization scale or
scheme ambiguity the effective charges of the polarized Bjorken and the
Gross-Llewellen Smith sum rules from the experimental value for the effective
charge associated with $R$-ratio. Present data are consistent with the
generalized Crewther relations, but measurements at higher precision and
energies will be needed to decisively test these fundamental relations in QCD.
