As far as I know, you can basically install how many desktop environments as you want, as long as they are available for your distribution -in fact, many distros come with more than one desktop environment by default (you usually can choose which one to use at the login screen), and you can always remove the one you don't want and/or add another one.

But then, removing a desktop environment almost always results in leaving behind stuff that can't be eliminated because it will break something, and adding another one on top of the default will likely result in this
There is always some clutter and application duplication after installing more than one desktop environment
as Frogs Hair pointed above.

So yes, I'd say you can probably install Cinnamon on Ubuntu-Mate (Mint had both MATE and Cinnamon installed a while ago), as long as you add the right Cinnamon repo for Ubuntu.

I originally asked if Ubuntu-Mate would be cleaner than a standard Ubuntu installation on which I add MATE, looks like it will probably be but I reckon I'll have to try a live session to see how different it is.