  The spin up of helium II is studied by calculating the spin-down recovery of
a superfluid-filled container after an impulsive acceleration and comparing
with experiments. The calculation takes advantage of a recently published
analytic solution for the spin up of a Hall-Vinen-Bekharevich-Khalatnikov
superfluid that treats the back-reaction torque exerted by the viscous
component self-consistently in arbitrary geometry for the first time. Excellent
agreement at the 0.5% level is obtained for experiments at $T=1.57\,{\rm K}$,
after correcting for the non-uniform rotation in the initial state, confirming
that vortex tension and pinning (which are omitted from the theory) play a
minimal role under certain conditions (small Rossby number, smooth walls). The
dependence of the spin-down time on temperature and the mass fraction of the
viscous component are also investigated. Closer to the lambda point, the
predicted onset of turbulence invalidates the linear Ekman theory.
