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 ...
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’,
Mullioned windows are windows divided into panes by inner frames. The vertical frames are called mullions. What do we call the horizontal frames?
I want to work with the terms horizontal/vertical, but there's no term I can find to describe the z axis. Furthermore, from what I've read, many people define the horizontal direction as being the horizontal and vertical direction, with height being described as the vertical direction i.e, horizontal = area, vertical = height.
(possible) interest only: I use || to separate distinct thoughts in a comment field such as this one. || Using a double vertical separator is exceedingly non-standard but I think/hope/feel conveys its intended meaning well.
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.)