Which sentence has the correct subject-verb agreement?
neither the western hemisphere nor the eastern hemisphere is immune to flooding.
neither the western hemispheres nor the eastern hemisphere are immune to flooding.

I think that the first sentence has the correct subject-verb agreement because it sounds right to me and if you take the sentence apart and say; The Western Hemisphere is immune to flooding or The Eastern Hemisphere is immune to flooding, the sentence makes sense. This is what I learned in my 6th grade english class. 

It’s the first one.