= Solution
Reflect the configuration in the horizontal plane through the cylinder axis. The wall and cylinder geometry are unchanged. Downward translation becomes upward translation, a possible wall-normal migration <velocity> is unchanged, and the axial angular <velocity> changes sign because it is an axial <vector>.
On the other hand, reversing the original <Stokes flow> makes all three <velocities> change sign: translation becomes upward, wall-normal migration reverses, and rotation reverses. The reflected and reversed problems therefore have the same forcing and geometry except that their wall-normal <velocities> have opposite signs. Uniqueness of <Stokes flow> forces that <velocity> to equal its own negative, so it is zero. The rotational <velocities> already agree under the two operations, so this argument places no restriction on rotation.
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