If I want to speak of North, South, East, West in a general sense I could, for example, use the term cardinal direction. Which term is appropriate to sum up horizontal and vertical in the same man...
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 ...
The convention is that x would occupy the horizontal axis, while y occupies the vertical axis, regardless if x is plotted against y, or y against x. Visually, which often would appear mutually indiscriminatable for 1-1 mapping plots.
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 ...
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’,
2 'Horizontal' means 'relating to the horizon', so strictly speaking whether a split is vertical or horizontal depends on its orientation relative to the ground. Or less strictly, 'horizontal' is whatever the observer considers to be left/right rather than up/down.