More information.... the orders still appear to EXIST, but are not accessible from the order overview page:
/wp-admin/edit.php?post_type=shop_order
Orders can not be searched for, either. Not by order #, name, or any other parameter. As far as the order overview page is concerned, they don't exist.
However, if I go directly to an order #, the information is all still there. For example:
/wp-admin/post.php?post=#######&action=edit
Obviously, this is less then ideal, though at least this didn't DELETE the orders, as that would be an epic cataclysm for us. We need to be able to search orders and see an overview of orders. Any ideas?