Most lattices can be partitioned into even and odd sublattices, much as the squares on a checkerboard can be colored red and black, for example. Sums can be taken separately in the two lattices, since the neighborhoods will reflect the same partitioning; but when there is a central cell, it can be considered separately, to preserve the balance between the two classes. Rules reflecting this partitioning are called even-odd-center rules.
Considering the neighborhood in isolation from the lattice, the concept of even and odd can be interpreted fairly liberally; for instance as the top of the neighborhood and the bottom of the neighborhood. Sometimes interesting variants on the rules result from such creative partitioning.
Displaying an even-odd-center rule requires two 
matrices
for a Moore neighborhood, because there are two classes with an
occupancy running between zero and four, modified by the two values
of the central cell.