  We extend the improved Collins-Soper-Sterman (iCSS) $W+Y$ construction
recently presented in~\cite{Collins:2016hqq} to the case of polarized
observables, where we focus in particular on the Sivers effect in
semi-inclusive deep-inelastic scattering. We further show how one recovers the
expected leading-order collinear twist-3 result from a (weighted)
$q_T$-integral of the differential cross section. We are also able to
demonstrate the validity of the well-known relation between the (TMD) Sivers
function and the (collinear twist-3) Qiu-Sterman function within the iCSS
framework. This relation allows for their interpretation as functions yielding
the average transverse momentum of unpolarized quarks in a transversely
polarized spin-$\frac{1}{2}$ target. We further outline how this study can be
generalized to other polarized quantities.
