  Dilation surfaces are generalizations of translation surfaces where the
geometric structure is modelled on the complex plane up to affine maps whose
linear part is real. They are the geometric framework to study suspensions of
affine interval exchange maps. However, though the $SL(2,\mathbb{R})$-action is
ergodic in connected components of strata of translation surfaces, there may be
mutually disjoint $SL(2,\mathbb{R})$-invariant open subsets in components of
dilation surfaces. A first distinction is between triangulable and
non-triangulable dilation surfaces. For non-triangulable surfaces, the action
of $SL(2,\mathbb{R})$ is somewhat trivial so the study can be focused on the
space of triangulable dilation surfaces.\newline In this paper, we introduce
the notion of horizon saddle connections in order to refine the distinction
between triangulable and non-triangulable dilation surfaces. We also introduce
the family of quasi-Hopf surfaces that can be triangulable but display the same
trivial behavior as non-triangulable surfaces. We prove that existence of
horizon saddle connections drastically restricts the Veech group a dilation
surface can have.
