[en] We examine the possibility that the double axial charge operator could have a significant effect on the rates for 2ν ββ-decay. Although the new ββ-decay operator is nominally suppressed by (v/c)², it is not subject to the Pauli effects that weaken allowed matrix elements. In standard shell model treatments of the double beta decay nuclei ⁷⁶Ge, ⁸²Se, and ¹³⁰Te, this operator contributes as a first-order correction. By using the schematic Nilsson pairing model, we argue that core-polarization contributions generated by the quadrupole-quadrupole interaction will systematically cancel against the dominant Gamow-Teller amplitude. The contribution from the tensor force may also be important, but has not yet been evaluated. 10 refs., 2 figs., 2 tabs