I know that "эта ту" (ta tu) (which is not an abbreviation, so ignore the periods in the English spelling) means "this one, that one", and is short for "Эта девушка любит ту девушку." ("This girl loves that girl.") I remember one of them was interviewed by phone on the radio, and this is how she explained it when asked what t.A.T.u. stood for.