Beginning with this result of 4x7 = 28 moves on the 8x8 board.This is clearly the best possible since the maximum for a closed tour without the symmetry condition is known to be 32. It would surprise me if this has not been published somewhere before, but no-one has so far found an earlier reference.
Next I found a 4x13 = 52 move tour on the 10x10 board. This is also probably the maximum since the best closed tour known takes 54 moves.
Finally the best known closed tour on the 12x12 board uses 86 moves so I tried to form a quaternary tour with 4x21 = 84 moves, but could only manage three examples using 4x19 = 76 moves.
Can anyone fit in the extra two moves in each quarter, or is that impossible?