  During accretion a neutron star (NS) is spun up as angular momentum is
transported through its surface layers. We study the resulting differentially
rotating profile, focusing on the impact this has for type I X-ray bursts. The
predominant viscosity is likely provided by the Tayler-Spruit dynamo. The
radial and azimuthal magnetic field components have strengths of ~10^5 G and
~10^10 G, respectively. This leads to nearly uniform rotation at the depths of
interest for X-ray bursts. A remaining small shear transmits the accreted
angular momentum inward to the NS interior. Though this shear gives little
viscous heating, it can trigger turbulent mixing. Detailed simulations will be
required to fully understand the consequences of mixing, but our models
illustrate some general features. Mixing has the greatest impact when the
buoyancy at the compositional discontinuity between accreted matter and ashes
is overcome. This occurs at high accretion rates, at low spin frequencies, or
may depend on the ashes from the previous burst. We then find two new regimes
of burning. The first is ignition in a layer containing a mixture of heavier
elements from the ashes. If ignition occurs at the base of the mixed layer,
recurrence times as short as ~5-30 minutes are possible. This may explain the
short recurrence time of some bursts, but incomplete burning is still needed to
explain these bursts' energetics. When mixing is sufficiently strong, a second
regime is found where accreted helium mixes deep enough to burn stably,
quenching X-ray bursts. We speculate that the observed change in X-ray burst
properties near one-tenth the Eddington accretion rate is from this mechanism.
The carbon-rich material produced by stable helium burning would be important
for triggering and fueling superbursts. (abridged)
