= Solution
Write $\mathbf n=\mathbf R/R$. At fixed geometry, <Linearity of Stokes flow> makes the translational velocities linear in the couples. With zero forces and couples perpendicular to $\mathbf R$, isotropy and reflection symmetry allow only terms proportional to $\mathbf G_j\times\mathbf n$ in each $\mathbf U_i$. There is no longitudinal term: reflect in the plane containing $\mathbf R$ and the common torque axis, taking account of the axial-vector transformation of torque, and use linearity under torque reversal. Hence
$$
\dot R=\mathbf n\cdot(\mathbf U_2-\mathbf U_1)=0,
\qquad\boxed{R\text{ is constant}.}
$$
The direction of $\mathbf R$ can still change; constant separation does not mean stationary centers.
When the couples are equal, a rotation through $\pi$ about their common axis exchanges the identical spheres and reverses the in-plane translational velocities while leaving the axial angular velocities unchanged. Uniqueness of <Stokes flow> then gives
$$
\boxed{\mathbf U_2=-\mathbf U_1,\qquad\boldsymbol\Omega_2=\boldsymbol\Omega_1.}
$$
The midpoint is stationary and the pair can orbit it. The body spin need not equal the angular rate of that orbit. These symmetry statements hold at arbitrary noncontact separation, not just in a distant <rotlet> approximation.
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