With your particular workbook, you may have some (limited) options. This is because the pills on your ROWS shelf aren't actually being used to establish a hierarchical structure; they're only there to ensure that a separate row is created for every possible combination of those fields. Your actual values are being generated as labels on stub SUM(0.0) columns.
Replace all the pills on the ROWS shelf with a single calculated field that combines all of the field values together. This calculated field should change the order in which the fields are combined, according to the sort that the user wants (e.g. create a parameter to gather the sort preference from the user).
For instance (simplifying):
[Combined sort field]
CASE [Sort parameter]
WHEN 'Product Name' THEN [Product Name] + [Region] + [Customer Name] + [Customer Segment] + STR(Order Quantity) + STR(Shipping Cost)
WHEN 'Region' THEN [Region] + [Product Name] + [Customer Name] + [Customer Segment] + STR(Order Quantity) + STR(Shipping Cost)
WHEN 'Customer Name' THEN [Customer Name] + [Product Name] + [Region] + [Customer Segment] + STR(Order Quantity) + STR(Shipping Cost)
// ... et cetera
(I apologize, I don't have time to build an example workbook right now, but try this out as an exercise to see if it will satisfy your requirements.)