Parity of sets of mutually orthogonal Latin squares

Let the ensemble of an OA be the set of Latin squares derived by interpreting any three columns of the OA as a Latin square. We demonstrate many restrictions on the number of Latin squares of each parity that the ensemble of an OA(k,n) can contain. These restrictions depend on n(mod4) and give some insight as to why it is harder to build projective planes of order n≡2(mod4) than for n≢2(mod4). For example, we prove that when n≡2(mod4) it is impossible to build an OA(n+1,n) for which all Latin squares in the ensemble are isotopic (equivalent to each other up to permutation of the rows, columns and symbols).