  Einstein--Cartan gravity is a close historical sibling of general relativity
that allows for spacetime torsion. As a result, angular momentum couples to
spacetime geometry in a similar way to energy. While consequences of this are
well studied on cosmological scales, their role in neutron star physics is
largely under-explored. We study the effects that torsion, sourced by either
microphysical spin or macroscopic angular momentum, has on neutron stars. For
this, we use a simplified polytropic model to quantify the microphysical
coupling to torsion. We also derive expressions to model rotation-induced
torsion effects and estimate the consequences for rotating neutron stars with
different rotation rates. We find that the presence of torsion in general leads
to neutron stars with smaller radii and masses, but higher central densities.
Realistic models for microphysical spin lead to torsion effects that have no
relevant influence on the neutron star structure. Rotation-induced torsion
effects however, can decrease the radius by up to $900\,m$, which is comparable
to the increase due to centrifugal forces. Depending on which effect dominates,
this leads to a torsion-induced spin-up or spin-down of the neutron star. We
conclude that torsion effects due to rotation can not be neglected and are
large enough to be tested using current or near-future technology.
