We characterise the quotient surface graphs arising from symmetric contact systems of line segments in the plane and also from symmetric pointed pseudotriangulations in the case where the group of symmetries is generated by a translation or a rotation of finite order. These results generalise well known results of Thomassen, in the case of line segments, and of Streinu and Haas et al., in the case of pseudotriangulations. Our main tool is a new inductive characterisation of the appropriate classes of surface graphs. We also discuss some consequences of our results in the area of geometric rigidity theory.