On wordpress.barrycarter.info, the links sidebar is in alphabetical order by category ("My ... Stuff"), which looks weird for "My other stuff". How can I order the link categories myself?
Well, the plugin My Link Order does exactly that.
This is in a widget? Or coded into the sidebar? http://codex.wordpress.org/Function_Reference/wp_list_bookmarks
is the code for showing your blogroll links, with orderby parameters.
If you are particular about the order of categories, and category_orderby isn't cutting it, you can use wp_list_bookmarks numerous times, each time showing only one category
If you are using the widget,maybe look for a different plugin with widget, or consider hardcoding this?