Hamming distance for conjugates

Let x, y be strings of equal length. The Hamming distance h(x, y) between x and y is the number of positions in which x and y differ. If x is a cyclic shift of y, we say x and y are conjugates. We consider f(x, y), the Hamming distance between the conjugates xy and yx. Over a binary alphabet f(x, y) is always even, and must satisfy a further technical condition. By contrast, over an alphabet of size 3 or greater, f(x, y) can take any value between 0 and |x|+ |y|, except 1; furthermore, we can always assume that the smaller string has only one type of letter.