When I right click the desktop, the context menu doesn’t come up. I may have inadvertently disabled it somehow. Is there any way I can turn it back on?
Go into MATE Tweak, and in the Desktop tab make sure Show Desktop Icons is checked.
Don't know why not showing desktop icons makes the right click menu go bye-bye. 
I suspect it’s because “show desktop icons” is actually a euphemism for “manage desktop”. One of the things involved in managing the desktop is showing desktop icons. But, another will be allowing context based menus to be used on it.
That fixed the issue. Thanks a lot! 
Indeed, that actually means Caja is managing the desktop, so it reacts to right-clicks as well. It's not just in mate-tweak, it's the same in the code and in Caja settings.
The naming isn't very good, I know. I think we inherited that from Gnome 2 when MATE was forked. Though I'm not really sure - I started contributing to MATE a few years later after it started. 
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Thanks for the explanation Monsta. It’s fine, really, since right-click menus can still be made available (by having “show desktop icons” enabled) whilst simultaneously not having any of the system icons on the desktop (by individually de-selecting each of those system icons). This is, indeed, how I set my own desktop up.