This is in Adobe Camera RAW and in Lightroom, the color temperature sliders are Blue/Yellow and Green/Magenta, why isn't there one for Red/Cyan?
The third is a result of the first two. You have to set a ratio for each color in relation to another. If you turn up both the blue and the green, this effectively reduces the red. Similarly, if you go all the way towards yellow and magenta, then you increase the red.
Put another way, green/magenta is the balance between green vs red and blue and blue/yellow is the balance between blue vs red and green. Using the two together can cancel out the green and blue effect of each other to shift the red relative to them.
They could just as easily choose any set of two of the three. I'm not sure if there is a particular reasons they choose red as the odd man out, though I'd guess it might be harder, perceptually, to white balance that way. We tend to do really well at recognizing blues (sky/water) and greens (grass/trees) where as red is less programmed in to us.
To answer why there's no Red/Cyan balance we need to understand why Green/Magenta and Blue/Yellow are used.
This is down to the CIE XYZ model based on the work of Wright/Guild back in 1931. These studies are direct studies into human colour perception and they now underpin pretty much all of colour theory.
In their research, Green is represented in the widest range of wavelengths in our field of vision. Magenta is the opposite colour.
Blue then represents the remaining range of wavelengths in our field of vision, again balanced by Yellow as the opposite. They are also 90 degrees from Green/Magenta in the colour wheel.
Between those two we can identify any colour on the colour wheel (and a bunch of imaginary colours besides) so having a 3rd set of control points for colour would be of no benefit.