I've been told that (conceptually) a (par) asset swap package is effectively a swap that is created by taking the fixed leg to be the bond cashflows and the floating leg has the same payment frequency except that it is based off LIBOR + some spread, so that altogether, it values to 1-P, where P is the price the bond is trading at. And this seems sensible enough... But what about the case where the bond is a zero coupon bond and therefore has only 1 cashflow at maturity? Then what on earth does the spread represent (seeing as there are no libor coupons in a matching floating leg!)Can someone please enlighten me? I really fail to understand what the asset swap spread measures in the case of a zero coupon bond.

You know, the asset swap for a bond (not zero coupon) don't match with the Libor. For example take a german bond(yearly frequency) and euribor (6-month frequency).

I think one counterparty will still have to pay periodic libor (+spread) without getting something in return. The spread is what makes the present value the Floating Leg equal to the Fixed Leg (no matter if it is coupon paying or zero).