  We critically compare thermodynamic and kinetic approaches, that have been
recently used to study relations between the spin polarization and fluid
vorticity in systems consisting of spin-one-half particles. The thermodynamic
approach refers to general properties of global thermal equilibrium with a
rigid-like rotation and demonstrates that the spin-polarization and
thermal-vorticity tensors are equal. On the other hand, the kinetic approach
uses the concept of the Wigner function and its semi-classical expansion. In
most of the works done so far, the Wigner functions satisfy kinetic equations
with a vanishing collision term. We show that this assumption restricts
significantly applicability of such frameworks and, in contrast to many claims
found in the literature, does not allow for drawing any conclusions regarding
the relation between the thermal-vorticity and spin-polarization tensors,
except for the fact that the two should be constant in global equilibrium. We
further show how the kinetic-theory equations including spin degrees of freedom
can be used to formulate a hydrodynamic framework for particles with spin. We
define hydrodynamic equations starting separately from the formulation by
de~Groot, van~Leeuwen, and van~Weert and from the canonical formalism. In the
former case the energy-momentum tensor is symmetric and the spin tensor is
conserved, while in the later case the energy-momentum tensor is not symmetric
and the spin tensor is not conserved. Nevertheless, in the two cases the total
angular momentum is always conserved. Interestingly, the two approaches are
connected by the pseudo-gauge transformation, which we explicitly define.
