  We study the polarized Bjorken sum rule at low momentum transfers in the
range $0.22<Q<1.73 {\rm GeV}$ with the four-loop N$^3$LO expression for the
coefficient function $C_{\rm Bj}(\alpha_s)$ in the framework of the common QCD
perturbation theory (PT) and the singularity-free analytic perturbation theory
(APT). The analysis of the PT series for $C_{\rm Bj}(\alpha_s)$ gives a hint to
its asymptotic nature manifesting itself in the region $Q<1$ GeV. It relates to
the observation that the accuracy of both the three- and four-loop PT
predictions happens to be at the same 10% level. On the other hand, the usage
of the two-loop APT allows one to describe the precise low energy JLab data
down to $Q\sim 300$ MeV and gives a possibility for reliable extraction of the
higher twist (HT) corrections. At the same time, above $Q\sim 700$ MeV the APT
two-loop order with HT is equivalent to the four-loop PT with HT compatible to
zero and is adequate to current accuracy of the data.
