  We present a combinatorial model for cluster algebras of type $D_n$ in terms
of centrally symmetric pseudotriangulations of a regular $2n$-gon with a small
disk in the centre. This model provides convenient and uniform interpretations
for clusters, cluster variables and their exchange relations, as well as for
quivers and their mutations. We also present a new combinatorial interpretation
of cluster variables in terms of perfect matchings of a graph after deleting
two of its vertices. This interpretation differs from known interpretations in
the literature. Its main feature, in contrast with other interpretations, is
that for a fixed initial cluster seed, one or two graphs serve for the
computation of all cluster variables. Finally, we discuss applications of our
model to polytopal realizations of type $D$ associahedra and connections to
subword complexes and $c$-cluster complexes.
