As several people have pointed out, your example has degree $0$. Another way to
see this is to observe that given a section, the number of zeros minus poles in both hemispheres is a difference of two winding numbers. This would work out to
$(1/2\pi i)\int_\gamma d\log g_{12}$, where $\gamma$ is the equator and $g_{12}$ is the
transition function (as in Henri's answer). In fancier terms, this is the
first Chern number. In your example, this works out to $0$ (again). For $\mathbb{P}^1$,
the degree is the sole invariant.