Stacking-Controlled Magnetic Exchange and Magnetoelectric Coupling in Bilayer CrI$_2$

We use a first-principles calculations approach to reveal the electronic and magnetic properties of chromium diiodide (CrI$_2$) bilayers and establish a hierarchy of magnetic interactions across stable registries. The monolayer presents a x-stripe antiferromagnetic ground state, while in bilayers the BA$^\prime$ stacking is the global minimum with antiparallel interlayer magnetic alignment. Bilayer configurations strengthen the exchange in the plane by 6 % to 10 %, while the exchange between layers is registry-dependent. The symmetry of each stacking configuration allows for anisotropic interactions. Dzyaloshinskii-Moriya terms appear in structures without inversion symmetry, which in this case also generates in-plane polarizations of up to $\sim$ 10 $\mu$C/cm$^2$, resulting in direct magnetoelectric coupling that is absent in centrosymmetric monolayers. Thus, stacking acts both as a selector of exchange anisotropy and as a driver of magnetoelectricity. Our results show that bilayer CrI$_2$ can be mechanically reconfigured through interlayer sliding, with energy differences between stacking orders (25-50 meV/f.u.) that are compatible with experimental actuation. Tunable magnetism and register-dependent polarization offer promising opportunities for novel spintronic devices, where structural transitions can affect both magnetic states and electric dipoles.