Explanation of the Marching Squares saddle points resolution
Imagine that a black dot represents a value of -1 whereas a white dot represents a value of +1. You are looking for the iso-line(s) where each point upon the line is 0.

For this example you could either place the iso-lines like on the left or the right:

 or 

The difference between the solutions is the value in the center. In the left case we assume 1s whereas in the right solution we assume -1s. So we check the center value and based on that value decide which solution fits best. If the center value is 1, we pick the left solution. If the center value is -1, we choose the right solution.