A preference ordering R is called "self-dual" by Samuelson if and only if there exists a direct utility function U representing R such that U(Z) = - U*(Z) is any non-negative n-vector and U* is the indirect utility function corresponding to U. Samuelson showed that the Cobb-Douglas preference ordering is self-dual and asked the open question as to the existence of any other self-dual case. If a preference ordering R is both self-dual and homothetic, then for the two-good case Samuelson claims to have proved that R is Cobb-Douglasian and conjectures the same to be true in the three-or-more good-case. Swamy has claimed that the Cobb-Douglas case is the only example of a preference ordering which is self-dual and either homothetic or additive. In this paper, we give two non Cobb-Douglasian examples of self-duality, one additive and the other homothetic, in order to answer the open question and refute the claims.