  The newly discovered two-dimensional magnetic semiconductors such as CrI$_3$
have triggered a surge of interest stemming from their exotic spin-dependent
properties and potential applications in spintronics and
magneto-optoelectronics. Using first-principle density-functional-theory
calculations, we investigate the properties of the spin-polarization texture in
momentum space in the prototype 2D centrosymmetric ferromagnetic (FM) bilayer
of CrI$_3$, with perpendicular magnetization. The FM bilayer displays a rich
in-plane spin texture in its highest valence bands. We show the existence of
two distinct spin canting effects, which result from the coupling of FM order
with structural chirality and electric polarization, in establishing the
in-plane spin texture. The first effect is generated by the chirality of the
layer stacking and the spin-orbit-polarized nature of the valence states, and
yields the same canting on both layers. The second effect is a Rashba-related
effect, which in a centrosymmetric ferromagnet induces in a single electronic
state two opposite spin-canted components on the two layers. Furthermore, using
the FM bilayer as an example, we provide general rules imposed by
magnetic-space-group symmetries on spin-polarization vectors, serving as
guidelines for the shape of the spin texture in the Brillouin zone of magnetic
crystals. Finally, we show that the above effects can be effectively used to
manipulate the spin texture via compressive strain, which induces in the FM
bilayer valence-band-edge states with canted spins.
