  In continuation of a preceding work on introducing asymmetric thin-shell
wormholes as an emerging class of traversable wormholes within the context,
this time cylindrically symmetric spacetimes are exploited to construct such
wormholes. Having established a generic formulation, first the Linet-Tian
metric generally, and then the cosmic string metric and a black string metric
in greater details are studied as constructing blocks of cylindrical asymmetric
thin-shell wormholes. The corresponding wormholes are investigated within the
linearized stability analysis framework to firstly, demonstrate that they can
exist from the mechanical stability point of view, and secondly, indicate the
correlation between the stability and symmetry in each case, if there is any at
all. From here, we have extracted a pattern for the way stability changes with
the asymmetry degree for the two examples; however, it was observed that the
symmetric state is not the most neither the less stable state. There are also
some side results: It was learned that any cylindrical thin-shell wormhole made
of two cosmic string universes cannot be supported by a barotropic equation of
state. Furthermore, as another side outcome, it was perceived that the radius
dependency of the so-called variable equation of state, which is used all over
this article, has a great impact on the mechanical stability of the cylindrical
asymmetric thin-shell wormholes studied in this brief.
