Order embeddings of real matrix domains

Let $n$ be a positive integer, $n≠1$, and $S_n$ the set of all $n\times n$ real symmetric matrices. A nonempty subset $\text{U}\subset S_n$ is called a matrix domain if it is open and connected and a map $\phi :\text{U}\to S_n$ is said to be an order emebedding if for every pair $X,Y\in \text{U}$ we have $X\le Y⟺\phi (X)\le \phi (Y)$. We describe the general form of such maps.