This sequence, by Euler, was found in Phreno-Mnemotechny by Fauvel Gouraud, 1845. It used sequential square numbering, instead of cartesian coordinates for the squares.

Any tour sequence can start & end on any square (w/ 65 steps).
It has 16 simple tropes: starting from any of the 4 corners, and stepping to the right or left.
(I did not count going in reverse sequence, nor starting elsewhere in the sequence.)
The center sortie can likewise be independently rotated, but not flipped, for 64 versions of this tour.
Other solutions exist. Can you find one of them? Do you know how to guess your odds of finding one?

In the following 2 tables, the steps are numbered in 2 sequences:
1) Stepwise 1 thru 64 as done by Gouraud.
2) Stepwise in octets from 11, 21, 31, ... That is 11-18, 21-28, ... -85.

Elsewhere, they are numbered in Octal coded Decimal for mnemonic purposes.

(1)
   col.	1	2	3	4	5	6	7	8  <-col.
row                                                                     row
1	1	52	15	34	3	50	17	36	1
2	14	33	2	51	16	35	4	49	2
3	53	64	31	24	29	26	37	18	3
4	32	13	62	27	60	23	48	5	4
5	63	54	11	30	25	28	19	38	5
6	12	43	56	61	22	59	6	47	6
7	55	10	41	58	45	8	39	20	7
8	42	57	44	9	40	21	46	7	8

(2)
   col.	1	2	3	4	5	6	7	8  <-col
row                                                                    row
1	11	71	27	47	13	67	31	51	1
2	25	46	12	68	28	48	14	66	2
3	72	85	44	38	42	42	52	32	3
4	45	25	83	40	81	37	65	15	4
5	84	73	23	43	41	41	33	53	5
6	24	58	75	82	36	78	16	64	6
7	74	22	56	77	62	18	54	34	7
8	57	76	61	21	55	35	63	17	8

	1	2	3	4	5	6	7	8