Finding all permutations which follow reflection symmetry

Question

I am unsure how to write down these questions formally, so here is some python code and the problem in words.

I am looking to find all permutations which subscribe to reflective symmetry. For example, given the permutation group $G$ over e.g. {1,2,3,4,5} then I am looking for e.g. {1,2,3,2,1} or {2,2,2,2,2} - i.e. those permutations which are reflective about the center of the permutation.

In code:

n = 6 # length of set
r = 4 # selection
a = [range(n)]*r
all_layouts = list(product(*a))

To find all symmetric permutations:

symmetric_layout = []
r2 = r//2
for t in all_motor_layouts:
    is_symmetric = True
    for i in range(r2):
        if t[i] != t[r - i - 1]:
            is_symmetric = False
            break
    if is_symmetric:
        symmetric_layout.append(t)

Examples of all_layouts:

[(0, 0, 0, 0),
 (0, 0, 0, 1),
 (0, 0, 0, 2),
 (0, 0, 0, 3),
 (0, 0, 0, 4),
 (0, 0, 0, 5),
 (0, 0, 1, 0),
 (0, 0, 1, 1),
 (0, 0, 1, 2),
 (0, 0, 1, 3),...]

Example of symmetric_layout:

[(0, 0, 0, 0),
 (0, 1, 1, 0),
 (0, 2, 2, 0),
 (0, 3, 3, 0),
 (0, 4, 4, 0),
 (0, 5, 5, 0),
 (1, 0, 0, 1),
 (1, 1, 1, 1),
 (1, 2, 2, 1),...]

Questions:

1. How is this problem more formally described? (palindrome permutations?) I.e. to find a sub-group of permutations which subscribe to some particular symmetry?
2. How can we find this group without, as above, enumerating all of them and seeing if they subscribe to our desired symmetry?

Thanks