= Solution
Apply the <divergence theorem> to $A(r)=(C\cdot r)(C'\cdot r)J(r)$. Since $C,C'$ are constant and $J$ is a <solenoidal vector field>, the <product rule> gives
$$
\nabla\cdot A=(C\cdot J)(C'\cdot r)+(C'\cdot J)(C\cdot r).
$$
Its boundary flux vanishes when $J$ has <compact support>, or when its normal component vanishes on the boundary of the integration volume. Integrating therefore proves
$$
\boxed{\int_V(C\cdot r)(C'\cdot J)\,dV=-\int_V(C'\cdot r)(C\cdot J)\,dV.}
$$
For a discontinuous field, the divergence-free hypothesis must include its boundary jump, in the sense of a <weak derivative>. A field that merely has zero <divergence> at interior points can still have a nonzero surface source; that weaker hypothesis would not justify discarding the flux. The result is the <antisymmetric first moment of a localized solenoidal field>.
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