Is there a single, more generic term that can be used to describe both a row and a column? In English, we can refer to a line as being horizontal or vertical, but unless we say ‘a line of something’,
To speak of a vertical row would seem somehow perverse. It would seem far more meaningful to speak of arranging things in a vertical line—to distinguish this line from some other possible line in a different orientation. (It might seem even more normal to speak of columns, but that is outside the scope of this Question.)
If x and y are horizontal, z is vertical; if x and z are horizontal, y is vertical. The words horizontal and vertical are generally used in a planar (2-dimensional) sense, not spatial (3-dimensional). Which is the reason you may not find a word corresponding to the third dimension along with horizontal and vertical.
single word requests - X, Y, Z — horizontal, vertical and ...
This would answer the original title of the question, which only specified synonyms for "horizontal" and "vertical" -- they're all directions. There can be many cells that are orthogonal, but not necessarily neighboring/touching.
Is there one word for both horizontal or vertical, but not diagonal ...
According to Wikipedia's architectural drawing page: A cross section, also simply called a section, represents a vertical plane cut through the object, in the same way as a floor plan is a horizontal section viewed from the top. This would suggest that section is only appropriate for vertical planes. However, section is more generally defined as, per dictionary.com: a representation of an ...