The list is here: https://wiki.archlinux.org/index.php/desktop_environment
This is the Arch wiki. You see a list of all desktops that are supported. The package name that pulls in the entire desktop is given after the || in the short description of each desktop, e.g. budgie-desktop for Budgie, cinnamon for the Cinnamon desktop.
Just locate that package in Add/Remove software, or install in the terminal using "sudo pacman -S <packagename>.
Mixing desktop environments will pull in a lot of duplicate software, file managers, etc. You cannot simply remove a desktop by uninstalling the <packagename> of the desktop. Thus, it may be wise to use the command line and note all packages that are pulled in. If you want to “undo” later, you then can remove all these packages.
You may break your system, so make sure that you have a good backup copy of your personal data at any time.