According to rules, is cricket a bowler's game or a batsman's game?

Kamindu Mendis from Sri Lanka bowls with left arm to right-handers, and right arm to left-handers

What if there is an ambidexterous batsman who wants to bat right for a right-arm bowler and left for a left-arm bowler?

And if both of them face each other, who will be given the first preference?