A preference ordering is self-dual if it has a direct utility function that can be written in the same mathematical form as the corresponding indirect utility function. Samuelson has raised the question whether there are any such preference orderings in addition to the trivial one discussed by him. It is shown here that there exists at least one family of self-dual preference orderings, of which the direct and indirect addilog preference orderings are limiting cases. Although various properties of the demand functions derived from self-dual addilog can be demonstrated, the mathematical form of the utility functions has not been determined.