Liability Swap
  
See: IOU. Cue the Barney theme song: "IOU, UOMe. We're a happy family..."
Yes, that's how it works in a liability swap. Your debts become mine; mine become yours.
A credit swap involves trading the proceeds from an investment in a debt obligation. You hold a 10-year bond paying a fixed rate of 5% a year. Meanwhile, your cousin has a 10-year floating-rate bond currently paying 4%, but with the ability to move around as general interest rates move. You make a trade. All the cash you earn from your bond goes to your cousin. All your cousin's proceeds go to you. You don't actually trade ownership of the bonds...just the amount earned from the investments.
Okay, so that situation represents a credit swap. A liability swap works under the same premise, only with money you owe instead of money you earn.
Imagine the other side of the credit swap situation. That 5% fixed rate bond was issued by Steady Payment Corp. Meanwhile, Changing Times Inc. issued the floating-rate bond your cousin bought. Those companies enter a liability swap. Now Steady Payment will pay the amount owed to your cousin and Changing Times will pay the amount owed to you. They are doing a criss-cross of who pays whom...a liability swap.