Symmetric transformation
Marios Mamzeris shows us that you can transform a odd square with sequencial numbers into a symmetric magic square in two steps (https://www.oddmagicsquares.com/):
Step 1, horizontal swap
| < | 1 | 2 | 3 | 4 | 5 | 2 | 3 | 4 | 5 | 1 | ||
| < | 6 | 7 | 8 | 9 | 10 | 8 | 9 | 10 | 6 | 7 | ||
| 11 | 12 | 13 | 14 | 15 | 11 | 12 | 13 | 14 | 15 | |||
| 16 | 17 | 18 | 19 | 20 | > | 19 | 20 | 16 | 17 | 18 | ||
| 21 | 22 | 23 | 24 | 25 | > | 25 | 21 | 22 | 23 | 24 |
Step 2, vertical swap
| ^ | ^ | |||||||||||
| 2 | 3 | 4 | 5 | 1 | 8 | 12 | 4 | 17 | 24 | |||
| 8 | 9 | 10 | 6 | 7 | 11 | 20 | 10 | 23 | 1 | |||
| 11 | 12 | 13 | 14 | 15 | 19 | 21 | 13 | 5 | 7 | |||
| 19 | 20 | 16 | 17 | 18 | 25 | 3 | 16 | 6 | 15 | |||
| 25 | 21 | 22 | 23 | 24 | 2 | 9 | 22 | 14 | 18 | |||
| v | v |
See 3x3, 5x5, 7x7, 9x9, 11x11, 13x13, 15x15, 17x17, 19x19, 21x21, 23x23, 25x25, 27x27, 29x29 and 31x31
