4.12 The acrobats are numbered $1, 2, \cdots$, 12. They are to be arranged in two circles, $A$ and $B$, each with six people. In circle $B$, each acrobat stands on the shoulders of two adjacent acrobats in circle $A$. If the number of each acrobat in circle $B$ is equal to the sum of the numbers of the two acrobats below him, then such an arrangement is called a "tower". How many structurally different towers can be formed?

[Note] Towers that are the same after rotation or reflection are considered the same structure. For example, with 8 people, draw a circle and fill in the numbers of the acrobats in the bottom layer inside the circle, and the numbers of the acrobats in the top layer outside the circle. In Figure 2, the three diagrams are all towers, but the last two can be obtained from the first one by rotation or reflection, so they belong to the same structure.
4. Let the sums of the elements in circles $A$ and $B$ be $x$ and $y$ respectively. Then $y=2x$. Therefore,
$$
3x = x + y = 1 + 2 + \cdots + 12 = 78.
$$

Solving for $x$ gives $x = 26$.
Clearly, $1, 2 \in A$ and $11, 12 \in B$.
Let $A = \{1, 2, a, b, c, d\} (a < b < c < d)$. Then $a + b + c + d = 23$, and $a \geq 3, 8 \leq d \leq 10$ (if $d \leq 7$, then $a + b + c + d \leq 4 + 5 + 6 + 7 = 22$, a contradiction).
(1) If $d = 8$, then
$$
A = \{1, 2, a, b, c, 8\}, c \leq 7, a + b + c = 15.
$$

Thus, $(a, b, c) = (3, 5, 7)$ or $(4, 5, 6)$, i.e.,
$$
A = \{1, 2, 3, 5, 7, 8\} \text{ or } \{1, 2, 4, 5, 6, 8\}.
$$

If $A = \{1, 2, 3, 5, 7, 8\}$, then
$$
B = \{4, 6, 9, 10, 11, 12\}.
$$

Since $B$ contains $4, 6, 11, 12$, in the $A$ circle, 1 must be adjacent to 3, 1 must be adjacent to 5, 5 must be adjacent to 7, and 8 must be adjacent to 3. In this case, there is only one arrangement, leading to one tower, as shown in Figure 6(a).
$$
\begin{array}{l}
\text{If } A = \{1, 2, 4, 5, 6, 8\}, \text{ then} \\
B = \{3, 7, 9, 10, 11, 12\}.
\end{array}
$$

Similarly, in the $A$ circle, 1 must be adjacent to 2, 5 must be adjacent to 6, and 4 must be adjacent to 8. In this case, there are two arrangements, leading to two towers, as shown in Figures 6(b) and (c).
(2) If $d = 9$, then
$$
A = \{1, 2, a, b, c, 9\}, c \leq 8, a + b + c = 14.
$$

Thus, $(a, b, c) = (3, 5, 6)$ or $(3, 4, 7)$, i.e.,
$$
A = \{1, 2, 3, 5, 6, 9\} \text{ or } \{1, 2, 3, 4, 7, 9\}.
$$

If $A = \{1, 2, 3, 5, 6, 9\}$, then
$$
B = \{4, 7, 8, 10, 11, 12\}.
$$

To get $4, 10, 12$ in the $B$ circle, in the $A$ circle, 1, 3, and 9 must be pairwise adjacent, which is impossible;
If $A = \{1, 2, 3, 4, 7, 9\}$, then
$$
B = \{5, 6, 8, 10, 11, 12\}.
$$

To get $6, 8, 12$ in the $B$ circle, in the $A$ circle, 2 must be adjacent to 4, 1 must be adjacent to 7, and 9 must be adjacent to 3. In this case, there are two arrangements, leading to two towers, as shown in Figure 7.
(3) If $d = 10$, then
$$
A = \{1, 2, a, b, c, 10\}, c \leq 9, a + b + c = 13.
$$

Thus, $(a, b, c) = (3, 4, 6)$, i.e.,
$$
A = \{1, 2, 3, 4, 6, 10\},
$$
$$
B = \{5, 7, 8, 9, 11, 12\}.
$$

To get $8, 9, 11, 12$ in the $B$ circle, in the $A$ circle, 6 must be adjacent to 2, 6 must be adjacent to 3, 10 must be adjacent to 1, and 10 must be adjacent to 2. In this case, there is only one arrangement, leading to one tower, as shown in Figure 8.

Therefore, there are 6 structurally distinct towers.