IMHO do NOT give your card for "auto debit" to pay any bills.
This is where the trouble lies. With the onset of bill paying on line it is so simple to just go on line and pay them monthly. This way 'you' have to enter all the personal info manually and after the transaction it is not stored on their server with ALL your info intact.
When I signed on for a sprint account the Sprint store strongly suggested I do NOT sign up for auto debit. That should tell you something!
UHaul rental is the biggest offender. I rented a truck. The card is swiped for the one time transaction, no info should be stored. Unknowingly to the public UHaul automatically takes all your card info and stores it available for anyone working at Uhaul to use it. In my case they did, which is how I discovered their practice.
I've been on the phone with them for 6 months, its built into their system. Even if you make the darn clerk delete it from the on line reciept.....you can go to the next UHAUL seven states away and they have it on file and don't even ask you for your card for your purchase, they push a button and your card is charged!!

As these hackers get better and criminal employees seeing how easy it is to steal, I see big problems coming down the pike with using debit cards and credit cards on line. Or in my case using it for any purchase from a company.:(