We consider magnetospheric structure of rotating neutron stars with internally twisted axisymmetric magnetic fields. The twist-induced and rotation-induced toroidal magnetic fields align/counter-align in different hemispheres. Using analytical and numerical calculations (with PHAEDRA code) we show that as a result the North-South symmetry is broken: the magnetosphere and the wind become angled, of conical shape. Angling of the magnetosphere affects the spindown (making it smaller for mild twists), makes the return current split unequally at the Y-point, produces anisotropic wind and linear acceleration that may dominate over gravitational acceleration in the Galactic potential and give a total kick up to $sim 100$ km/s. We also consider analytically the structure of the Y-point in the twisted magnetosphere, and provide estimate of the internal twist beyond which no stable solutions exist: over-twisted magnetospheres must produce plasma ejection events.
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