Here's something strange: seeing as how this seems to be panel-related, I had a prod inside all of the items in the top and bottom panels. When I examined the system tray widget, I found, inside system tray settings, some widgets that didn't appear in the system tray (I had been playing around with widgets a few days ago but thought I'd removed them all and cleaned up my desktop). The ones of interest are easy SSH connection, weather forecast, and bubblemon as shown below:
Now I know from previous experience with easy SSH connection, that it gave me nothing but grief (it ran once I installed/reinstalled and configured it, but every time I rebooted the machine, it would show up as broken (and when this happened, it took out the pager widget as well, making it fall over badly), so I decided to toss it.
So what I did was check the boxes under "extra items" to make the widgets show, closed the system tray settings window, right-clicked on these widgets to configure them, got them showing nicely (except for easy SSH connection, which remained showing as a big red box with an "X" in it saying "I give up"), and then right-clicked to "remove this..." to delete them from the system tray. They disappeared from the system tray, but when I go back into system tray settings, their entries are still there.
I suspect this might be what's causing the problem, especially since easy SSH connection always ends up breaking the pager widget. So... how do I get rid of these things permanently, in case their presence is the cause of this problem?