  All known solutions to the Einstein equations describing rotating cylindrical
wormholes lack asymptotic flatness and therefore cannot describe wormhole
entrances as local objects in our Universe. To overcome this difficulty,
wormhole solutions are joined to flat asymptotic regions at some surfaces
$\Sigma_-$ and $\Sigma_+$. The whole configuration thus consists of three
regions, the internal one containing a wormhole throat, and two flat external
ones, considered in rotating reference frames. Using a special kind of
anisotropic fluid respecting the Weak Energy Condition (WEC) as a source of
gravity in the internal region, we show that the parameters of this
configuration can be chosen in such a way that matter on both junction surfaces
$\Sigma_-$ and $\Sigma_+$ also respects the WEC. Closed timelike curves are
shown to be absent by construction in the whole configuration. It seems to be
the first example of regular twice asymptotically flat wormholes without exotic
matter and without closed timelike curves, obtained in general relativity.
