Is there one word for both horizontal or vertical, but not diagonal, adjacency?
I've seen both "Orthogonally Adjacent" (Adjacent at right angles) and "Edge Adjacent" (Adjacent across edges rather than corners) used. They do mean slightly different things, but it's only relevant when dealing with something other than a regular rectangular grid.
Since measuring distance in this kind of topology is called "Manhattan distance" you might also try "Manhattan adjacent" although I've never seen that usage myself and it would probably require some explanation before using it.
Consider Orthogonal
Orthogonal - intersecting or lying at right angles