Spherical varieties and p-adic families of cohomology classes

We prove a "twist-compatibility" result for p-adic families of cohomology classes associated to symmetric spaces. This shows that a single family of classes (lying in a finitely-generated Iwasawa module) interpolates classical cohomology classes of many different weights, including twists by Grossencharacters of possibly non-trivial infinity-type. This subsumes and generalises a number of prior results relating to Euler systems and p-adic L-functions, and we conclude with some novel applications to Euler systems for GSp(4), GSp(4) x GL(2), and GSp(4) x GL(2) x GL(2).