Symmetric Rhombic Tours

In November I looked at the problem of enumerating the symmetric tours of squares and diamonds type.  I found there were four distinct ways (1, 2, 3, 4) of arranging the diamonds in a quarter board, and twenty ways these could be arranged together on the board, namely in the ten pairs 11, 22, 33, 44, 12, 13, 14, 23, 24, 34, each either in direct (=) or oblique (~) symmetry. This enumeration is not yet complete but I found unique solutions in the 22= and 44~ cases, both being double halfboard tours.