These levers no longer remain attached to the door they are supposed to be on.
Originally I thought that they are supposed to be held onto the door by the dents in the rosettes that fit into the mounting brackets on the door.
However, I found the manufacturer's installation instructions for a similar model (PDF, see page 8) which shows that there is a bumpy portion of the spindle. Closer inspection reveals that the spindle itself has a square profile and the bumps are an additional piece of metal that is attached to the side of the spindle. (Note the metal hooks that come around the edge and into the hole.)
The problem here appears to be that somehow the bumps on the near end of the spindle have been pushed down far enough that they do not grab the inside of the handle.
I've been looking online, but the only spindles I can find are threaded ones, not this type of pressure-fit. I've tried various search terms, with no results.
What should I be searching for to find a replacement for this spindle?
UPDATE
Since there has been some discussion about how this mechanism actually attaches, here are some more pictures (along with what I hope is a better explanation).
First, here is a picture showing how the handle+rosette are attached to the door:
The red arrow points to one of the four notches in the plastic mounting bracket on the door. Four "ears" on the metal part of the rosette snap into these notches. (If you look at the close-up of the spindle above, you can see the ears in the background, though they are a little blurry.)
The blue and orange circles indicate which piece in the pictures corresponds to each piece in the manual linked above. The "rosette" circled in orange is a metal piece with a plastic decorative trim around it.
Here is a picture of the outside handle along with the instruction page showing disasembly:
(You can see that one screw is missing from the plastic mounting bracket.)
If you refer back to the previous picture, you can see that in frame "A1" of the instructions, it says to insert the spindle into one of the handles. Note that it specifically calls out placement of the ridged part. Also note that the instructions show mounting the spindle into one handle and then passing it through the mortise lock and placing the other handle on the other end. However, in the demounting instructions, it shows the spindle remaining in one of the handles:
