  We find a vacuum stationary twisted solution in four-dimensional Einstein
gravity. Its frame dragging angular velocities are antisymmetric with respect
to the equatorial plane. It possesses a symmetry of joint inversion of time and
parity with respect to the equatorial plane. Its Arnowitt-Deser-Misner (ADM)
mass and angular momentum are zero. It is curved but regular all over the
manifold. Its Komar mass and Komar angular momentum are also zero. Its infinite
red-shift surface coincides with its event horizon, since the event horizon
does not rotate. Furthermore we extend this solution to the massive case, and
find some similar properties. This solution is a stationary axisymmetric
solution, but not Kerr. It explicitly proves that pure Einstein gravity permits
different rotational mode other than Kerr. Our results demonstrate that the
Einstein theory may have much more rich structures than what we ever imagine.
