  Given a generic Looijenga pair $(Y,D)$ together with a toric model
$\rho:(Y,D)\rightarrow(\overline{Y},\overline{D})$, one can construct a seed
${\bf s}$ such that the corresponding $\mathcal{X}$-cluster variety
$\mathcal{X}_{{\bf s}}$ can be viewed as the universal family of the log
Calabi-Yau surface $U=Y\setminus D$. In cases where $(Y,D)$ is positive and
$\mathcal{X}_{{\bf s}}$ is not acyclic, we describe the action of the Weyl
group of $(Y,D)$ on the scattering diagram $\mathfrak{D}_{{\bf s}}$. Moreover,
we show that there is a Weyl group element ${\bf w}$ of order $2$ that either
agrees with or approximates the Donaldson-Thomas transformation ${\rm
DT}_{\mathcal{X}_{{\bf s}}}$ of $\mathcal{X}_{{\bf s}}$. As a corollary, ${\rm
DT}_{\mathcal{X}_{{\bf s}}}$ is cluster. In positive non-acyclic cases, we also
apply the folding technique as developed in \cite{YZ} and construct a maximally
folded new seed $\overline{{\bf s}}$ from ${\bf s}$. The $\mathcal{X}$-cluster
variety $\mathcal{X}_{\overline{{\bf s}}}$ is a locally closed subvariety of
$\mathcal{X}_{{\bf s}}$ and corresponds to the maximally degenerate subfamily
in the universal family. We show that the action of the special Weyl group
element ${\bf w}$ on $\mathfrak{D}_{{\bf s}}$ descends to
$\mathfrak{D}_{\overline{{\bf s}}}$ and permutes distinct subfans in
$\mathfrak{D}_{\overline{{\bf s}}}$ , generalizing the well-known case of the
Markov quiver.
