= Solution
Apply the <Kauffman bracket> skein relation to every crossing in the two-string <tangle> $D_1$. Each complete smoothing is a collection of closed circles together with one of the two crossingless pairings of the four boundary points. Rotation through $180^\circ$ about the indicated vertical axis preserves both crossingless pairings and the number of closed circles. Gluing the unchanged tangle $D_2$ to corresponding smoothings therefore gives equal terms, including equal loop factors, so $\langle D\rangle=\langle D'\rangle$.
Choose an orientation of $L$, and transport the induced orientations of the four ends of $D_1$ through the rotation to orient $L'$. Crossing signs inside the rotated tangle and outside it then have the same total, so the <writhe of a link diagram> satisfies $w(D)=w(D')$. Applying the writhe normalization of the <Jones polynomial> gives
$$
\boxed{V_L(t)=V_{L'}(t).}
$$
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