I've just completed an extensive piece of work collecting and constructing symmetric knight's tours on the normal chessboard. There are 183 tours on the page, showing all possible central angle formations.
Here is one group of tours from the page, (hh), (hi), (hj):
It was only a month or so ago that I was able to see the work by Brede 1844 that contains 24 tours, all but #14 being in 180 degree rotational symmetry. It has become available due to digitisation. This was evidently an important pioneering work that seems to have been under-appreciated. The tours are at the end of the book:
There is also a semimagic tour formed of two copies of a 4x8 semimagic tour joined together.