There are many parts to getting this right.

+ Some changes were necessary in Xmodmap.
+ A few options need to be specified.
+ The Maté settings daemon had to be disabled.

I tried switching layouts between the ones I use: US QWERTY and Greek.
The mod3 setup flawlessly now.

Concerning the settings daemon, I still need to find a solution, because
it provides some nice functionality I wish to keep.  Perhaps a more
lightweight daemon could be used here.  Will need to investigate.