Diagonal-complete latin squares

We call a latin square A=(aij) of order n, aij∈{1,2,…,n}, right-diagonal-complete if {(aij,ai+1,j+1):1≤i,j≤n}={(i,j):1≤i,j≤n} where the indices are periodic mod n. Left-diagonal-completeness is defined similarly. A latin square is called diagonal-complete if it is right- and left-diagonal-complete. It is shown that diagonal-complete latin squares of order n=4m always exist; for n=4m+2 no diagonal-complete-latin squares based on a group exist. For n∈{21,25,27,49,81} we found diagonal-complete latin squares by computer search. This search also showed that for n=9 there exist right-diagonal-complete latin squares but no diagonal-complete latin squares based on a group. For n∈{2,3,5,6,7}, there is no latin square which is right-diagonal-complete or left-diagonal-complete.