Yes the C T M - I agree, having the Mnemonic is nice, but you might as well use that as a reminder only, and understand the concept.
Why it happens like it does.
The formula for Dynamic Pressure, Mach and LSS are the ones that form the whole concept of the Chicked Tikka Masala rule!
CAS = Dynamic pressure = 1/2 r V2 (Vsquared) the V is TAS, r is density based on ISA conditions.
You go up, density decreases, so to keep the Dynamic pressure constant, TAS must increase.
This because you climb at a constant CAS.
So if you now climb at a constant TAS, because your density decreases, CAS must decrease.
However the reason Mach increases, is another formula and concept you should know.
LSS depends on temperature only, decreasing temperature, lower LSS. With altitude unless an inversion, LSS will get lower and lower until you reach the Tropopause.
Mach = TAS / LSS, so if TAS is constant, and your LSS is getting smaller the higher you fly, then Mach will increase.
It's all based on simple maths and common sense, if you apply that thought process, than it will make it much easier to explain if you get the question.
One thing is getting the question on a multiple choice, and you have all the time in the world, different the day somebody asks you, what happens to TAS when you climb at constant Mach? You will know the initial answer, however the follow up will be, ok, now explain me why?
Understanding the concept, makes it much easier to understand!
Ok, last one, I did not explain, but in brief, climb at constant Mach, your LSS will decrease with altitude, so to keep Mach constant, TAS also have to decrease with altitude, as you can see from the equation. (LSS = Local Speed of Sound)