【Example 2】Using 4 different colors to color the 4 sides of a small square wooden block, with each side painted a different color, how many different coloring patterns can there be?
Here, we not only consider the relative order of the 4 colors, but also the fact that the blocks can be flipped, so there is no distinction between clockwise and counterclockwise relative orders. Therefore, there are
$$
\frac{1}{2 \times 4} A_{4}^{4}=\frac{4 \times 3 \times 2}{2 \times 4}=3
$$

different coloring patterns. They are: (1) Color $A$ is opposite to Color $B$, and Color $C$ is opposite to Color $D$; (2) Color $A$ is opposite to Color $C$, and Color $B$ is opposite to Color $D$; (3) Color $A$ is opposite to Color $D$, and Color $B$ is opposite to Color $C$.