Punctured line bundles do not determine their duality sign

For any complex line bundle $L$, a nonzero dual vector $\varphi \in L_b^{\ast }$ determines the unique $v\in L_b$ with $\varphi (v)=1$. In local coordinates this is fiber inversion $a\mapsto a^{-1}$, giving a smooth projection-preserving diffeomorphism between the complements of the zero sections of $L^{\ast }$ and $L$. It is not a fiber-linear bundle isomorphism. In particular this condition cannot distinguish the tautological line on the complex projective line from its dual.