There is an infinite number of useful color axes. Well, nearly infinite. For example, think of Lab, with one lightness channel and two colour channels, and the choice of colour channels is fairly arbitrary. From an image encoded in Lab, we could convert to LCH, then rotate the hue, and convert back to Lab, so the axes would be different. We could have chosen to make one axis significant: a colour we want to desaturate or whatever. After tweaking, we have to convert to LCH, rotate the hue in the opposite direction, and convert to whatever colorspace we need for viewing.
I show an different example at Photo post / audio analogy - de-yellow night shot - #22 by snibgo, using simple 3x3 matrices to convert to/from sRGB, and one colour axis is from the hue of the cat’s fur.