  Neutron stars are born out of core-collapse supernovae, and they are imparted
natal kicks at birth as a consequence of asymmetric ejection of matter and
possibly neutrinos. Unless the force resulting from the kicks is exerted
exactly at their center, it will also cause the neutron star to rotate. In this
paper, we discuss the possibility that neutron stars may receive off-center
natal kicks at birth, which imprint a natal rotation. In this scenario, the
observed pulsar spin and transverse velocity in the Galaxy are expected to
correlate. We develop a model of the natal rotation imparted to neutron stars
and constrain it by the observed population of pulsars in our Galaxy. When
considering a single-kick position parameter, we find that the location of the
off-center kick is $R_{\rm kick}=2.03^{+3.74}_{-1.69}$\,km at $90\%$
confidence, and is robust when considering pulsars with different observed
periods, transverse velocities, and ages. Nonetheless, the model encounters
challenges in effectively fitting the data, particularly at small transverse
velocities, prompting the exploration of alternative models that include more
complex physics. Our framework could be used as a guide for core-collapse
simulations of massive stars.
