electric circuits - Capacitor charging, work done by battery on changing polarity

A uncharged capacitor $C$ is connected to a battery with potential $V$. It becomes fully charged and has a charge $Q=CV$ stored on it.

Now the polarity of the battery is reversed. The capacitor will have the charge $Q$ still but with polarity reversed too.

My question is: What is the work done by the the battery?

According to me, during 1st charging it will do a work of $QV$. The energy of the capacitor is also not changing, then what is the work done to change its polarity?