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.