  A combinatorial polytope $P$ is said to be projectively unique if it has a
single realization up to projective transformations. Projective uniqueness is a
geometrically compelling property but is difficult to verify. In this paper, we
merge two approaches to projective uniqueness in the literature. One is
primarily geometric and is due to McMullen, who showed that certain natural
operations on polytopes preserve projective uniqueness. The other is more
algebraic and is due to Gouveia, Macchia, Thomas, and Wiebe. They use certain
ideals associated to a polytope to verify a property called graphicality that
implies projective uniqueness. In this paper, we show that that McMullen's
operations preserve not only projective uniquness but also graphicality. As an
application, we show that large families of order polytopes are graphic and
thus projectively unique.
