We study the reflexivity of a Segre product of a projective space P-m and a projective variety Y, and give a criterion for P-m x Y to be reflexive in terms of m, the dimension of Y, the rank of the general Hessian of Y and the characteristic of the ground field. Our study is closely related to a question raised by Kleiman and Piene on the relationship between the conormal map and the Gauss map.