  Disk-fed accretion onto neutron stars can power a wide range of astrophysical
sources ranging from X-ray binaries, to accretion powered millisecond pulsars,
ultra-luminous X-ray sources, and gamma-ray bursts. A crucial parameter
controlling the gas-magnetosphere interaction is the strength of the stellar
dipole. In addition, coherent X-ray pulsations in many neutron star systems
indicate that the star's dipole moment is oblique relative to its rotation
axis. Therefore, it is critical to systematically explore the 2D parameter
space of the star's magnetic field strength and obliquity, which is what this
work does, for the first time, in the framework of 3D general-relativistic
magnetohydrodynamics. If the accretion disk carries its own vertical magnetic
field, this introduces an additional factor: the relative polarity of the disk
and stellar magnetic fields. We find that depending on the strength of the
stellar dipole and the star-disk relative polarity, the neutron star's jet
power can either increase or decrease with increasing obliquity. For weak
dipole strength (equivalently, high accretion rate), the parallel polarity
results in a positive correlation between jet power and obliquity, whereas the
anti-parallel orientation displays the opposite trend. For stronger dipoles,
the relative polarity effect disappears, and jet power always decreases with
increasing obliquity. The influence of the relative polarity gradually
disappears as obliquity increases. Highly oblique pulsars tend to have an
increased magnetospheric radius, a lower mass accretion rate, and enter the
propeller regime at lower magnetic moments than aligned stars.
