We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition $ξ$ with analogous normalized linear characters evaluated on the double partition $D(ξ)$. We also relate some natural filtration on the usual (linear) Kerov-Olshanski algebra of polynomial functions on the set of Young diagrams with its spin counterpart. Finally, we give a spin counterpart to Stanley formula for the characters of the symmetric groups.