  A graphene bilayer shows an unusual magnetoelectric response whose magnitude
is controlled by the valley-isospin density, making it possible to link
magnetoelectric behavior to valleytronics. Complementary to previous studies,
we consider the effect of static homogeneous electric and magnetic fields that
are oriented parallel to the bilayer's plane. Starting from a tight-binding
description and using quasi-degenerate perturbation theory, the low-energy
Hamiltonian is derived including all relevant magnetoelectric terms whose
prefactors are expressed in terms of tight-binding parameters. We confirm the
existence of an expected axion-type pseudoscalar term, which turns out to have
the same sign and about twice the magnitude of the previously obtained
out-of-plane counterpart. Additionally, small anisotropic corrections to the
magnetoelectric tensor are found that are fundamentally related to the skew
interlayer hopping parameter $\gamma_4$. We discuss possible ways to identify
magnetoelectric effects by distinctive features in the optical conductivity.
