  We demonstrate that rotating matter sets the throat of an Ellis wormhole into
rotation, allowing for wormholes which possess full reflection symmetry with
respect to the two asymptotically flat spacetime regions. We analyze the
properties of this new type of rotating wormholes and show that the wormhole
geometry can change from a single throat to a double throat configuration. We
further discuss the ergoregions and the lightring structure of these wormholes.
