I suspect that $$\iiint d \tau \,\,\, {\bf r} {\bf \cdot} {\bf c} \,\,\, {\bf J} {\bf \cdot} {\bf d}$$ is equal to $$\iiint d \tau \,\,\, \frac{1}{2} {\bf r} \times {\bf J} \, {\bf \cdot}\, {\bf c} \times {\bf d}$$ where ${\bf c}, {\bf d}$ are constant vectors and ${\bf J}$ is such that any surface integrals over all space involving ${\bf J}$ vanish.
I'm having a very difficult time showing this result and any help, hints would be appreciated.