The easiest solution would be to set a cookie. Then check what cookie the user have before you print the menu.
Another solution would be to add a query variable so that the two halves have different urls, for example mysite.com/?half=A and mysite.com/?half=B. Then you check the query var before you print the menu. This solution would make it possible with permalinks for the two different halves. (and you can make the urls pretty if you want)