  The class of traveling wave solutions of the sine-Gordon equation is known to
be in 1-1 correspondence with the class of (necessarily singular)
pseudospherical surfaces in Euclidean space with screw-motion symmetry: the
pseudospherical helicoids. We explicitly describe all pseudospherical helicoids
in terms of elliptic functions. This solves a problem posed by Popov
[Lobachevsky geometry and modern nonlinear problems, Birkh\"auser/Springer,
Cham, 2014]. As an application, countably many continuous families of
topologically embedded pseudospherical helicoids are constructed. A (singular)
pseudospherical helicoid is proved to be either a dense subset of a region
bounded by two coaxial cylinders, a topologically immersed cylinder with
helical self-intersections, or a topologically embedded cylinder with helical
singularities, called for short a pseudospherical twisted column.
Pseudospherical twisted columns are characterized by four phenomenological
invariants: the helicity $\eta\in \mathbb{Z}_2$, the parity $\epsilon\in
\mathbb{Z}_2$, the wave number $\mathfrak n\in \mathbb{N}$, and the aspect
ratio $\mathfrak{d}>0$, up to translations along the screw axis. A systematic
procedure for explicitly determining all pseudospherical twisted columns from
the invariants is provided.
