These diagrams are the results of a month's struggle to find a knight's tour with 90 degree rotational symmetry with the square numbers in a knight path (the red line). The tour on the circular board was completed first. Both tours use the same central pattern of knight moves. Both boards are of 22x22 = 484 cells. The yellow, blue and green paths are rotations of the red path. The circled cells mark where the coloured paths meet. These are at the points numbered 121, 242, 363, 484.
These are the only solutions to the quaternary Onitiu problem I have found so far. Whether it is possible on smaller boards (of sides 10, 14, 18) is unknown.