Let $X$ be a space and suppose:
For every two points $x,y\in X$ there is a homeomorphism $h$ that maps $X$ onto itself and such that $h(x)=y$ and $h(y)=x$.
Is there a name for this property?
Obviously $X$ is homogeneous. Does every homogeneous space have this property? If not, are there some classes of homogeneous spaces which have the property? For instance does every homogeneous compact connected metric space?
Such spaces are called bi-homogeneous (or perhaps bihomogeneous).
An example of a homogeneous space which is not bi-homogeneous is the long-line without the initial point (i.e., $X = ( \omega_1 \times [0,1) ) \setminus \{ ( 0 , 0 ) \}$). It is fairly easy to show that $X$ is homoegeneous, but since all homeomorphisms $X \to X$ are order-preserving it cannot be bi-homogeneous.
Many of the homogeneous spaces you encounter "in the wild" will be bi-homogeneous, but there doesn't appear to be many classes of topological spaces where homogeneity implies bi-homogeneity. One class of spaces for which this implication is (non-vacuously) true is the class of topological groups.
Homogeneous but not bi-homogeneous continua have been constructed. For one (and I believe the first) example see