Examining com.apple.notificationcenterui.plist showed that the names of the widgets that were missing from Notifications were also missing from that plist.
Removing that plist from /Users/username/Library/Preferences (obviously substituting my username), followed by logout and login, did indeed restore the missing widgets (they now appear in that plist).
To my great surprise, Notifications were still set up pretty much as they were—I expected them to be reset to default, but they were not. So apparently that plist doesn't determine what shows or in what order; instead, it determines what widgets are available to be shown or not shown. Cheers,
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Gib Henry