The authors study sets of three orthogonal partitioned incomplete Latin squares, of type \(2^ n\) (n odd), which have the property that two of the squares are mutual transposes and the third is symmetric. The main result is that such a set of 3 squares exists for odd \(n>3\), except possibly for n-15, 33, 39, 75 or 87. The result is proved by use of direct and recursive constructions including a PBD-closure result which is of interest in its own right: If \(P_ 5\) denotes the set of odd prime powers not less than 5, then there is a pairwise balanced design on v points with block sizes in \(P_ 5\), for all odd \(v>3\) except possibly for \(v=15\), 33, 39, 51, 75, 87, 93, 183, 195 or 219.