  We show that the intrinsic angular momentum of matter in curved spacetime
requires the metric-affine formulation of gravity, in which the antisymmetric
part of the affine connection (the torsion tensor) is not constrained to be
zero but is a variable in the principle of stationary action. Regarding the
tetrad and spin connection (or the metric and torsion tensors) as independent
variables gives the correct generalization of the conservation law for the
total (orbital plus intrinsic) angular momentum to the presence of the
gravitational field. The metric-affine formulation extends general relativity
to the simplest theory of gravity with intrinsic spin: the
Einstein-Cartan-Sciama-Kibble theory. We also show that teleparallel gravity,
which constrains the connection by setting the curvature tensor to zero, is
inconsistent with the conservation of the total angular momentum.
