the mapping w=-1/z corresponds to a 180 degree rotation of the Riemann sphere about the x2-axis (the imaginary axis).
The mapping $\displaystyle z\mapsto -\tfrac{1}{z}$ can be visualized as two mappings, the first $\displaystyle z\mapsto -z$ followed by $\displaystyle z\mapsto \tfrac{1}{z}$. Remember that $\displaystyle z\mapsto -z$ will rotate the Riemann sphere by 180 degrees while $\displaystyle z\to \tfrac{1}{z}$ will flip the Riemann sphere over upside down. Putting these effects together we get exactly what $\displaystyle z\mapsto -\tfrac{1}{z}$ does.