 Solution

For a closed path avoiding zero, choose a continuous lift $\gamma (t)/|\gamma (t)|=e^{i\theta (t)}$ and set $w(\gamma ;0)=(\theta (1)-\theta (0))/(2\pi )$. Conjugation replaces $\theta$ by $-\theta$, as does reversing the path, so both winding numbers are $-w(\gamma ;0)$.