Logical. Tested.
I had to use temporaries at one point, but there is only one solution.

See "My Challenge - Part 1" to learn more about this.

My conclusion:
If the numbers at the top are perfectly symmetrical (first and last columns have the exact same numbers in the exact same order, etc.) AND the numbers on the left are ALL palindromic (they are the same forwards as backwards, such as 3553 or 767), then there are 3 options:
1. The puzzle is NOT logical and not symmetrical (see "My Challenge - Part 1"). This type of puzzle will have multiple solutions.
2. The puzzle is NOT logical, but has only one solution. The solution MUST be symmetrical.
3. The puzzle IS logical AND it is also symmetrical.

In options 2 and 3, being symmetrical is NOT an assumption, it is a certainty. Therefore it is a logical deduction.

Reasoning:
Difficult to explain, but if you play Fqirytqles' "Playtime (7)":
https://www.hanjie-star.com/picross/playtime-7--25195.html
The bit at the end where there is a block of 3 and some other numbers that lead off it: there is only one solution, which is the symmetrical solution.
That exact same principle applies here, just on the scale of the whole puzzle.

The trick is to count very carefully: make sure that the numbers really are symmetrical/palindromic.
If they are, then either the puzzle is symmetrical or there are multiple solutions.
If they are not, then it is not logical to assume any symmetry.

I hope that makes sense.