Order embeddings of real matrix domains

Let $n$ be a positive integer, $n \not=1$, and $S_n$ the set of all $n \times n$ real symmetric matrices. A nonempty subset $\U \subset S_n$ is called a matrix domain if it is open and connected and a map $\phi : \U \to S_n$ is said to be an order emebedding if for every pair $X,Y \in \U$ we have $X \le Y \iff \phi (X) \le \phi(Y)$. We describe the general form of such maps.