  We present a new analysis of $A_N$ in $p^\uparrow p\to \pi\, X$ within the
collinear twist-3 factorization formalism. We incorporate recently derived
Lorentz invariance relations into our calculation and focus on input from the
kinematical twist-3 functions, which are weighted integrals of transverse
momentum dependent (TMD) functions. In particular, we use the latest
extractions of the Sivers and Collins functions with TMD evolution to compute
certain terms in $A_N$. Consequently, we are able to constrain the remaining
contributions from the lesser known dynamical twist-3 correlators.
