  In the paper the notion of a star partial homeomorphism of a finite
dimensional Euclidean space $\mathbb{R}^n$ is introduced. We describe the
structure of the semigroup $\mathbf{PStH}_{\mathbb{R}^n}$ of star partial
homeomorphisms of the space $\mathbb{R}^n.$ The structure of the band of
$\mathbf{PStH}_{\mathbb{R}^n}$ and Green's relations on
$\mathbf{PStH}_{\mathbb{R}^n}$ are described. We show that
$\mathbf{PStH}_{\mathbb{R}^n}$ is a bisimple inverse semigroup and every
non-unit congruence on $\mathbf{PStH}_{\mathbb{R}^n}$ is a group congruence.
