  We present a detailed theoretical study of bilayer-graphene's electronic
properties in the presence of electric and magnetic fields. Using
group-theoretical methods, we derive an invariant expansion of the Hamiltonian
for electron states near the K point of the Brillouin zone. In contrast to
known materials, including single-layer graphene, any possible coupling of
physical quantities to components of the external electric (magnetic) field has
a counterpart where the analogous component of the magnetic (electric) field
couples to exactly the same combination of quantities. For example, a purely
electric spin splitting appears as the magneto-electric analogue of the
familiar magnetic Zeeman spin splitting. The measurable thermodynamic response
induced by magnetic and electric fields is thus completely symmetric. The Pauli
magnetization induced by a magnetic field takes exactly the same functional
form as the polarization induced by an electric field. Our findings thus reveal
unconventional behavior of spin and pseudospin degrees of freedom in their
coupling to external fields. We explain how these counterintuitive couplings
are consistent with fundamental principles such as time reversal symmetry. For
example, only a magnetic field can give rise to a macroscopic spin
polarization, whereas only a perpendicular electric field can induce a
macroscopic polarization of the sublattice-related pseudospin degree of freedom
characterizing the intravalley orbital motion in bilayer graphene. These rules
enforced by symmetry for the matter-field interactions clarify the nature of
spins versus pseudospins. We also provide numerical values of prefactors for
relevant coupling terms. While our theoretical arguments use bilayer graphene
as an example, they are generally valid for any material with similar
symmetries.
