Was reading through the man page for fluxbox to look for something I spotted in the ~/.fluxbox/init file (the fluxbox resource file) and I came across this:
session.titlebar.{left|right}: buttons
The buttons or icons to place in the titlebar of decorated windows.
You may specify any number, space-delimited.
The available options are:
Close Maximize MenuIcon Minimize Shade Stick LHalf RHalf
Default left: Stick
Default right: Shade Minimize Maximize Close
LHalf and RHalf are buttons to quickly place a window into the left
and right half of the current monitor.
I never knew abouit the LHalf and RHalf buttons before; I've seen lhalf.pixmap and rhalf.pixmap settings in themes before, but thought it was a reference to the window.
So I checked them out, and sure enough it's a button to tile left/right on the screen:

Unfortunately, they do not toggle like the Maximize, Shade, and Stick button do so the use is limited. Still, and interesting feature of fluxbox.