Classification of symplectic surfaces and confusion about "symplectomorphism"

Question

So I read that the unique invariant of symplectic surfaces is the total area, i.e. two surfaces are symplectomorphic iff their area is the same. Consider $S^2$ with polar coordinates $(h,\theta)$ wherever these exist and the symplectic forms $\omega_1=dh\wedge d\theta$ and $\omega_2=-dh\wedge d\theta$. These should be non-symplectomorphic as the areas are not the same. But the reflection $\phi:(h,\theta)\rightarrow (-h,\theta)$ gives $\phi^*(\omega_1)=\omega_2$. What am I missing here?