Can you reuse Priority Mail boxes?
Yes, for Priority Mail. DMM 601.6.1a says USPS-produced packaging must be used only for the applicable product, so the same carton going out again on Priority Mail postage is doing exactly what it was made for.
Can I ship Ground Advantage in a Priority Mail box?
No. The container sets the price, not the label. Under DMM 601.6.1b the mailer of matter sent in USPS-produced packaging must pay the appropriate product price, which leaves you owing the Priority Mail figure on a cheaper label.
Does turning the box inside out fix it?
No, and the manual names that case. DMM 601.6.1b applies regardless of how the packaging is reconfigured or how markings may be obliterated, which takes in inside-out cartons, taped-over printing and marker.
Can a Priority Mail box go out with UPS or FedEx?
No. It is postal supply issued for a postal product under 601.6.1a, and no other network has any claim on it. Plain corrugated of the same size keeps every carrier on the table.
What happens if I send one anyway?
The obligation does not go away. Under 601.6.1b the piece still owes the Priority Mail price, and a shortfall has to be settled before anyone hands the parcel over, so expect a delay rather than a bargain.