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In reaction of $\ce{Al2O3}$ with $\ce{NaOH}$, we get product written as "$\ce{NaAlO2}$", "$\ce{Na2Al2O4}$" or just "$\ce{Na2O.Al2O3}$".

My question is what's the real structure of this compound? Spinel or a salt of aluminate anion ($\ce{AlO2-}$)? Maybe just mixture?

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From wikipedia article of sodium aluminate:

Anhydrous sodium aluminate, $\ce{NaAlO2}$, contains a three-dimensional framework of corner linked $\ce{AlO4}$ tetrahedra. The hydrated form $\ce{NaAlO2·5/4H2O}$ has layers of $\ce{AlO4}$ tetrahedra joined into rings and the layers are held together by sodium ions and water molecules that hydrogen bond to O atoms in the $\ce{AlO4}$ tetrahedra.

The composition of sodium aluminate is basically written as $\ce{NaAlO2, NaAl(OH)4}$ (hydrated), $\ce{Na2O·Al2O3}$ or $\ce{Na2Al2O4}$. 

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