When you are searching are you searching under aliases?  If so, check the alias of one of your org's and I bet there is an entry starting with 'The'.

"Within RE" if you have the slash in the Org name you should be able to sort on Org Name and it wil only use what is after the slash.  There is not a separate field.

However, if you export the data to excel or the like you lose the backslash functionality altogether on org name.  I think that either Sort Name or Sort Key might work but I have had trouble with this in the past and can't guarantee that either work.

It's not an alias.  I’ve checked every field in both the Constituent and Records tables but no field contains what I'm looking for.  Can you think of any other tables to try? Do you think BB has hidden the table?  (It's so frustrating to be able to do it "within RE" but not without.)

I've just tested it and the Sort Key field should work for you.

It is possible!  (Thanks to Biran McMickle, Developer Solutions Specialist.)  There is a view called SearchOrg that contains a field called Key_Name.  Key_Name contains the Organization name, but if there is a slash (\) in the name it only contains the characters after the slash.  (SearchOrg.Records_ID links to Records.ID.)  I just tested it and now I can get perfectly sorted lists!

Yeah!