You can user other Desktop environments other than default MATE Desktop. There are several ways as below:
Use "Live task". There are several live tasks for Desktop environments as below provided by Debian.
live-task-cinnamon --> Cinnamon Desktop
live-task-gnome --> GNOME Desktop
live-task-kde --> KDE Desktop
live-task-lxde --> LXDE Desktop
live-task-lxqt --> LXQt Desktop
live-task-xfce --> Xfce Desktop
For example, run the following command to install cinnamon:
sudo apt install live-task-cinnamon
Use Tasksel
This is a typical Debian way to install Desktop.
Run the command below as root from MATE Terminal:
apt install tasksel
Then, run tasksel as root
tasksel
As for Live Desktops based on MX Linux, use MX PakageInstaller --> Desktop Environments
It will show a list of Desktops: Budgie, GNOME, KDE, LXDE, Sugar, etc.
So that, select one you like and click the button for automatic installation.