A tautonym was originally the same word repeated for genus and species, though in Word Ways it is used of a single word consisting of two identical halves, such as Wallawalla, a North American Indian people (OED). I take it that it is not necessary for the two halves to have the same meaning, is for the semantic split to occur also in the middle. Suppose we start a 10-letter word square with a tautonym ABCDEABCDE. The second row must begin with B, say BFGHI (we will just use the first half for brevity), the third with CG, eg CGJKL, and so on.
Word Ways: Vol. 36
, Article 20.
Available at: http://digitalcommons.butler.edu/wordways/vol36/iss1/20