When a homogeneous convex cone is given, a natural partial order is introduced in the cone. We shall show that a homogeneous convex cone is a symmetric cone if and only if Vinberg´s -map and its inverse reverse the order. Actually our theorem is formulated in terms of the family of pseudoinverse maps including the -map, and states that the above order-reversing property is typical of the -map of a symmetric cone which coincides with the inverse map of the Jordan algebra associated with the symmetric cone.
Chifune KAI. "A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps." J. Math. Soc. Japan 60 (4) 1107 - 1134, October, 2008. https://doi.org/10.2969/jmsj/06041107