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At a flat shear-free impermeable plane, a tangential torque is reflected with opposite sign because torque is an axial vector. A parallel axial rotlet dipole therefore has an opposite signed moment at the reflected source. Its velocity vanishes at the point directly beneath the image, but its velocity gradient can turn the swimmer.

The rotlet is the point-torque fundamental solution of Stokes flow. Its torque strength $\mathbf{R}$ is an axial vector, and its circulating velocity decays as $r^{-2}$. A freely swimming cell with no external torque is torque-free. The motor applies opposite internal torques to its body and flagellar apparatus, so the net torque monopole satisfiesIndividual body and flagellar torques need not vanish. A spatially separated pair can therefore leave a rotlet dipole even though there is no far-field rotlet monopole.

A flagellar bundle and the counterrotating body supply oppositely signed axial torques separated along the swimming direction. Their leading torque moments cancel, and the next term is a rotlet dipole, decaying as $r^{-3}$. For the ideal axisymmetric bacterium the torque vector is parallel or antiparallel to $\mathbf{e}$.

To fix the sign convention, define a signed dipole strength $D$ with units torque times length and setDifferentiating givesMore generally, differentiating $\mathbf{R}\times \mathbf{r}/r^3$ gives $\mathbf{R}\times \mathbf{e}/r^3-3(\mathbf{e}\cdot \mathbf{r})(\mathbf{R}\times \mathbf{r})/r^5$; the first term vanishes when $\mathbf{R}\parallel \mathbf{e}$. A derivative with respect to source position instead of observation position changes the definition of the signed moment. The physical handedness depends on which torque is ahead, so it cannot be fixed without that convention.

For an axial rotlet dipole parallel to a flat free surface, its reflected image gives the displayed normal vorticity at height $h$. A spherical torque-free body responds with angular velocity ${\omega }_z/2$ and an elongated body also responds to strain. At constant height, self-propulsion plus yaw gives a circular path; the signed moment sets its handedness.