  Pulsars may either be spun up or down by hydrodynamic instabilities during
the supernova explosion of massive stars. Besides rapidly-rotating cases
related to bipolar explosions, stellar rotation may affect the explosion of
massive stars in the more common situations where the centrifugal force is
minor. Using 2D simulations of a simplified setup in cylindrical geometry, we
examine the impact of rotation on the Standing Accretion Shock Instability
(SASI) and the corotation instability, also known as low-T/|W|. The influence
of rotation on the saturation amplitude of these instabilities depends on the
specific angular momentum in the accretion flow and the ratio of the shock to
the neutron star radii. The spiral mode of SASI becomes more vigorous with
faster rotation only if this ratio is large enough. A corotation instability
develops at large rotation rates and impacts the dynamics more dramatically,
leading to a strong one-armed spiral wave. Non-axisymmetric instabilities are
able to redistribute angular momentum radially and affect the pulsar spin at
birth. A systematic study of the relationship between the core rotation period
of the progenitor and the initial pulsar spin is performed. Stellar rotation
rates for which pulsars are spun up or down by SASI are estimated. Rapidly
spinning progenitors are modestly spun down by spiral modes, less than $\sim
30\%$, when a corotation instability develops. Given the observational
constraints on pulsar spin periods at birth, this suggests that rapid rotation
might not play a significant hydrodynamic role in most core-collapse
supernovae.
