Example 2 Fill the numbers $1,2, \cdots, 9$ into the squares of a $3 \times 3$ chessboard (as shown in Figure 1), so that the sum of the absolute values of the differences between numbers in adjacent (sharing a common edge) squares is maximized. Find this maximum value.
Solution: To avoid the concept of absolute value, we now define each difference as the larger number minus the smaller number.

To maximize the sum of the 12 differences, we hope each difference is as large as possible, which means the minuend should be as large as possible and the subtrahend as small as possible.

Notice that, the center cell in Figure 1 has four adjacent cells, so the center cell should be filled with the largest number 9, and the four adjacent cells should be filled with the smaller four numbers 1, 2, 3, 4 (as shown in Figure 4), to make these 4 differences as large as possible. The remaining four numbers 5, 6, 7, 8 are then placed in the four corners of Figure 4. According to the rule mentioned in Example 1, the sum of the 8 differences formed by the perimeter of Figure 4 is also maximized.

Therefore, the maximum value of the sum of the absolute values of these 9 differences is
$$
\begin{array}{l}
9 \times 4+(8+7+6+5) \times 2-(1+2+3+4) \times 3 \\
=58 .
\end{array}
$$