Originally Posted by Yllanes

It depends. In the case of your itinerary the JFK-GIG one-way ticket on AA costs X avios + 5 $ (where X depends on the booked cabin, of course). So if you cancel it more than 24 hours before departure you get your avios back and the only thing you lose is your $ 5. They do not charge a "redeposit" fee to get your avios back like some other airlines. If the booking fee is less than the $ 55 cancellation fee you only lose what you already paid, not more.

For other itineraries, where you might need to pay hundreds of dollars on top of your avios, it is cheaper to cancel if they are all in one booking, like you say.