  We study the cohomology theory and the canonical Milnor-Witt cycle module
associated to a motivic spectrum. We prove that the heart of Morel-Voevodsky
stable homotopy category over a perfect field (equipped with its homotopy
t-structure) is equivalent to the category of Milnor-Witt cycle modules, thus
generalising D\'eglise's thesis. As a corollary, we recover a theorem of
Ananyevskiy and Neshitov and we prove that the Milnor-Witt K-theory groups are
birational invariants.
