13. (15 points) Eight $2 \times 1$ small rectangles as shown in the figure can be used to form a $4 \times 4$ square. If a square formed by these small rectangles is the same as another square after rotation, they are considered the same square. How many different squares can be formed that are symmetrical top-to-bottom, have two adjacent empty small squares in the first row, and are considered the same if they can be rotated to match another?

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13. (15 points) Eight $2 \times 1$ small rectangles as shown in the figure can be used to form a $4 \times 4$ square. If a square formed by these small rectangles is the same as another square after rotation, they are considered the same square. How many different squares can be formed that are symmetrical top-to-bottom, have two adjacent empty small squares in the first row, and are considered the same if they can be rotated to match another?
【Analysis】Use the right figure
to replace the $2 \times 1$ rectangle in the problem, because the given small rectangle is asymmetric up and down, so the same small rectangle in the square composed of it will not be in both the upper and lower parts at the same time. Therefore, we only need to consider different cases in the upper part, and analyze and solve the problem accordingly.

【Solution】Solution: (1) The adjacent blank cells are at the far left or far right of the first row. Since we need to exclude the same after rotation, we only consider the case where the adjacent blank cells are on the far right, as shown in the following two cases:
(2) The adjacent blank cells are in the middle of the first row. After removing the cases that overlap after rotation, there are 3 cases as shown in the following figures:
Answer: In all possible square patterns, there are 5 patterns that are symmetrical up and down, have two blank cells in the first row, and the blank cells are adjacent.