  Two related constructions are studied: (1) The diagonal complex $\mathcal{D}$
and its barycentric subdivision $\mathcal{BD}$ related to a \textit{punctured}
oriented surface $F$ equipped with a number of labeled marked points. (2) The
symmetric diagonal complex $\mathcal{D}^{inv}$ and its barycentric subdivision
$\mathcal{BD}^{inv}$ related to a symmetric (=with an involution) oriented
surface $F$ equipped with a number of (symmetrically placed) labeled marked
points. Eliminating a puncture gives rise to a bundle whose fibers are
homeomorphic to a surgery of the surface $F$. The bundle can be viewed as the
"universal curve with holes". The symmetric complex is shown to be homotopy
equivalent to the complex of a punctured surface obtained by a surgery of the
initial symmetric surface.
