Orientable homotopy modules

We prove a conjecture of Morel identifying Voevodsky’s homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy $t$-structure and have a trivial action of the Hopf map. This is done by relating these two kind of objects to Rost’s cycle modules. Applications to algebraic cobordism and construction of cycle classes are given.