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Let $B_p$ be the Latin square given by the addition table for the integersmodulo an odd prime $p$. Here we consider the properties of Latin trades in$B_p$ which preserve orthogonality with one of the $p-1$ MOLS given by thefinite field construction. We show that for certain choices of the orthogonalmate, there is a lower bound logarithmic in $p$ for the number of times eachsymbol occurs in such a trade, with an overall lower bound of$(\log{p})^2/\log\log{p}$ for the size of such a trade. Such trades imply theexistence of orthomorphisms of the cyclic group which differ from a linearorthomorphism by a small amount. We also show that any transversal in $B_p$hits the main diagonal either $p$ or at most $p-\log_2{p}-1$ times. Finally, if$p\equiv 1\mod{6}$ we show the existence of Latin square containing a $2\times2$ subsquare which is orthogonal to $B_p$.

Orthogonal trades in complete sets of MOLS