  We consider Ellis wormholes immersed in rotating matter in the form of an
ordinary complex boson field. The resulting wormholes may possess full
reflection symmetry with respect to the two asymptotically flat spacetime
regions. However, there arise also wormhole solutions where the reflection
symmetry is broken. The latter always appear in pairs. We analyse the
properties of these rotating wormholes and show that their geometry may feature
single throats or double throats. We also discuss the ergoregions and the
lightring structure of these wormholes.
