Is balanced Hamiltonian cycle NP complete on maximal plane graphs? I know that the Hamiltonian cycle is NP complete on the class of maximal plane graphs. 

If we instead ask about balanced Hamiltonian cycles (i.e. same number of faces on both sides) on maximal plane graphs, is the problem still NP complete?

My intuition is that this extra requirement shouldn't make the problem easier, but maybe this feeling is naive since after all it means that the reduction has to be more intricate.
Yes, it is still $NP$ complete. This is because of:

Claim: All Hamiltonian cycles on maximal planar graphs are balanced.

Proof: This is a special case of Grinberg's theorem: https://en.wikipedia.org/wiki/Grinberg%27s_theorem