  We consider a symbolic coding for geodesics on the family of Veech surfaces
(translation surfaces rich with affine symmetries) recently discovered by Bouw
and Moeller. These surfaces, as noticed by Hooper, can be realized by cutting
and pasting a collection of semi-regular polygons. We characterize the set of
symbolic sequences (cutting sequences) that arise by coding linear trajectories
by the sequence of polygon sides crossed. We provide a full characterization
for the closure of the set of cutting sequences, in the spirit of the classical
characterization of Sturmian sequences and the recent characterization of
Smillie-Ulcigrai of cutting sequences of linear trajectories on regular
polygons. The characterization is in terms of a system of finitely many
substitutions (also known as an S-adic presentation), governed by a
one-dimensional continued fraction-like map. As in the Sturmian and regular
polygon case, the characterization is based on renormalization and the
definition of a suitable combinatorial derivation operator. One of the
novelties is that derivation is done in two steps, without directly using Veech
group elements, but by exploiting an affine diffeomorphism that maps a
Bouw-Moeller surface to the dual Bouw-M\"oller surface in the same Teichmueller
disk. As a technical tool, we crucially exploit the presentation of
Bouw-M\"oller surfaces via Hooper diagrams.
