Why we can not store AC in batteries like DC?

In batteries, a direct current causes the electron exchanging chemical reaction to shift its equilibrium such that one terminal gets surplus of electrons and a shortage of electrons in the other thereby storing potential energy between terminals. 

When these terminals are connected to an external electrical load, they release the energy by exchanging electrons and because this exchange is unidirectional, we see it as DC. there is no method which can cause this motion to naturally move these electrons back and forth and that's why we don't hear about storing AC. 

When we try to charge battery by AC, As the battery terminals are fixed, the battery will charge in the positive half of the voltage and will be discharged in the negative half of the voltage. At the end there will be zero charge in the battery. So we can not charge a battery with AC.