  We consider stationary, cylindrically symmetric configurations in general
relativity and formulate necessary conditions for the existence of rotating
cylindrical wormholes. It is shown that in a comoving reference frame the
rotational part of the gravitational field is separated from its static part
and forms an effective stress-energy tensor with exotic properties, which
favors the existence of wormhole throats. Exact vacuum and scalar-vacuum
solutions (with a massless scalar) are considered as examples, and it turns out
that even vacuum solutions can be of wormhole nature. However, solutions
obtainable in this manner cannot have well-behaved asymptotic regions, which
excludes the existence of wormhole entrances appearing as local objects in our
Universe. To overcome this difficulty, we try to build configurations with flat
asymptotic regions by the cut-and-paste procedure: on both sides of the throat,
a wormhole solution is matched to a properly chosen region of flat space at
some surfaces $\Sigma_-$ and $\Sigma_+$. It is shown, however, that if we
describe the throat region with vacuum or scalar-vacuum solutions, one or both
thin shells appearing on $\Sigma_-$ and $\Sigma_+$ inevitably violate the null
energy condition. In other words, although rotating wormhole solutions are
easily found without exotic matter, such matter is still necessary for
obtaining asymptotic flatness.
