Thanks, all.
I guess I wasn't completely clear about the issue. For example, Callender says "...pinned items can be re-ordered. I'm not sure why that doesn't work for you." It does work for me. I can re-order to my heart's content. My issue is the need to reorder, as the taskbar shortcuts' order do not persist. I'm simply wondering if there's something I can do to eliminate that need. These programs are pinned. (The messages of course are not. I use Windows Live Mail, although I hope eventually to find a better replacement—it's the world's worst email program. I'd use Outlook except that I can't afford and can't otherwise use Microsoft Office.)
I believe that the changes take place upon system restart. A screen shot will only show the shortcuts where they are now (which is where I want them).
I didn't know Quick Launch was still available in Win7, in view of the pinning capability. I'll look into that.