After an agent finishes making changes, I've had success with:

"Can you give me a minute while I check my reservation online?"

While the agent is waiting on the phone, you open the reservation on United.com and check that it looks correct. You may also want to use "Add email address and request receipt"; the "View Receipt" link often doesn't update with the new eTicket until you force it to generate a new one by email. That part takes more time, though, so you may want to open the eTicket in the Saudia website. Note that the eTicket number doesn't always change when you change your ticket.

By the way, it's a good idea to stay out of your reservation while the agent is working on it; I've had agents tell me that they got an error when making changes because the record is locked (why on earth do they let the read-only website view of the record do this?).