= Solution
Choose the handedness for which the helix tangent is $\mathbf t=\cos\theta\,\mathbf e_z+\sin\theta\,\mathbf e_\phi$. Its local rigid velocity is $\mathbf u=\widetilde U\mathbf e_z+R\widetilde\Omega\mathbf e_\phi$, so $\mathbf u\mathbin\cdot\mathbf t=\widetilde U\cos\theta+R\widetilde\Omega\sin\theta$. Integrating the axial force and torque from the drag law gives the symmetric <hydrodynamic resistance matrix> in the question, with
$$
\boxed{B=(c_\parallel-c_\perp)RL\sin\theta\cos\theta.}
$$
Reversing helical handedness reverses the sign of $B$ but leaves $A$ and $D$ unchanged.
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