= Solution
Reflection through the projection plane reverses the orientation of $S^3$ and exchanges the overpassing and underpassing strands at every crossing. Thus switching all crossings gives the <mirror of a link>, with the component orientations transported by the reflection.
Define $W(L)(q)=V(\overline L)(q^{-1})$. Reflection exchanges $D_-$ with $D_+$ and preserves the <oriented smoothing>. Apply the printed <skein relation> to the reflected triple at parameter $q^{-1}$:
$$
q^{-1}W(D_+)-qW(D_-)=(q^{-1/2}-q^{1/2})W(D_0).
$$
Multiplication by $-1$ gives the original <skein relation> for $W$. Also $W(U_1)=1$. The stated uniqueness of the <Jones polynomial> now gives $W=V$, hence
$$
\boxed{V(\overline L)(q)=V(L)(q^{-1}).}
$$
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