  Given a map $\mathcal M$ on a connected and closed orientable surface, the
delta-matroid of $\mathcal M$ is a combinatorial object associated to $\mathcal
M$ which captures some topological information of the embedding. We explore how
delta-matroids associated to dessins d'enfants behave under the action of the
absolute Galois group. Twists of delta-matroids are considered as well; they
correspond to the recently introduced operation of partial duality of maps.
Furthermore, we prove that every map has a partial dual defined over its field
of moduli. A relationship between dessins, partial duals and tropical curves
arising from the cartography groups of dessins is observed as well.
