The problem boils down to finding the number of partitions of the set $\{1,2,\dots,n\}$ up to symmetry (i.e. $1\mapsto n$, $2\mapsto n-1$, ...), let's call this number $K_n$. First of all, the number of all partitions is the $n$-th Bell number $B_n$. From there we should be able to count the partitions up to symmetry using Burnside's lemma as
$$K_n=\frac 1 2 \left(B_n + F_n\right),$$
where $F_n$ is the number of partitions that are invariant under the symmetry. Counting these is more difficult than I imagined, when I made my comment earlier.
To tackle the problem, I wrote a Sage script to find $K_n$ and $F_n$:
B = []
K = []
F = []
for n in range(1,8):
partitions = SetPartitions(n).list()
B_n = len(partitions)
for p in partitions:
flipped = p.apply_permutation(Permutations(n).identity().reverse())
if p != flipped:
partitions.remove(flipped)
K_n = len(partitions)
F_n = 2*K_n-B_n
B.append(B_n)
K.append(K_n)
F.append(F_n)
The results are
$$
\begin{align*}
B_n &= (1, 2, 5, 15, 52, 203, 877, \dots),\\
K_n &= (1, 2, 4, 11, 32, 117, 468, \dots),\\
F_n &= (1, 2, 3, 7, 12, 31, 59, \dots).
\end{align*}
$$
Plugging the sequences into OEIS we find
- A000110: Bell numbers,
- A103293: Number of ways to color n regions arranged in a line such that consecutive regions do not have the same color,
- A080107: Number of fixed points of permutation of SetPartitions under $\{1,2,\dots,n\}\to\{n,n-1,\dots,1\}$.
Exactly what we were looking for. The formular given for $F_n$ on OEIS involes $q$-analog Bell numbers, which I haven't heard of before.
