Specific Ad-ID codes are not listed on the invoices.
Purchases, and therefore invoices, are not directly linked to specific codes. A purchase can be made and then codes created against that purchase at any time afterward. So in many cases, the invoice is generated before a code is even created.
When a user makes a purchase, they have the option to include an invoice note that will display on the invoice. If they know the code, such as those who are entering custom codes, they could enter it into the invoice note at the time of purchase. Alternatively, the user could edit the invoice and add the codes to the invoice note.