These are images that I put on twitter, partly stimulated by work by Robert Bosch.
They show ways of combining repetitions of a symmetric tour in an array 2x2, 3x3, 4x4 to form a single larger symmetric tour. The odd case simply uses a series of Vandermonde style links (two deletions and two insertions). The even cases use a linkage octagon (four deletions and four insertions) around the centre point, and the centre of each quarter in the 4x4 case.
The tours used are from Wenzelides 1849 and Haldeman 1864.