We report our findings with theoretical arguments on chiral symmetry breaking on the surface of charged cylinders. We use a model for periodic patterns of charges constrained over a cylindrical surface. In particular we focus on patterns of oriented lamellar patterns, such as, chiral helices, achiral rings or vertical lamellae, with the constraint of global electroneutrality. We study the dependence of the patterns' size and pitch angle on the radius of the cylinder and salt concentration. We obtain a phase diagram by using numerical and analytic techniques. For pure Coulomb interactions, we find a ring phase for small radii and a chiral helical phase for larger radii. We extend the findings to discrete triangular lattices wrapped over a cylindrical geometry. We find no symmetry breaking chiral helical phase in the discrete wrapping when using just an electrostatic potential and the minimum energy configuration is an achiral lattice matching the six-fold symmetry of triangular lattice. Conversely, with the addition of an elastic potential between the charges on the surface of the cylinder we find a stable chiral configuration. We discuss possible consequences and generalizations of our model.