Despite considerable efforts I have not made much further progress on the Quaternary Onitiu problem, that is of constructing knight tours with 90 degree rotational symmetry and with the square numbers in a knight path. These diagrams show the best I have found on the 10x10 and 14x14 boards in the form of Emperor tours which use wazir moves as well as knight moves.
I am now convinced that a knight tour solution on the 10x10 is impossible, and probably also on the 14x14 board, though I don't have a simple conclusive proof. In the above diagrams the square numbers are shown on the red line. The other coloured lines are rotations of the red line.
ADDENDUM (20 July 2021)
I have now solved the same problem on the 18x18 board, using only two wazir moves in each quarter. Can this be modified to make a knight tour solution?