  We use polarized inelastic neutron scattering to study the spin-excitations
anisotropy in the bilayer iron-based superconductor CaKFe$_4$As$_4$ ($T_c$ = 35
K). In the superconducting state, both odd and even $L-$modulations of spin
resonance have been observed in our previous unpolarized neutron scattering
experiments (T. Xie {\it et al.} Phys. Rev. Lett. {\bf 120}, 267003 (2018)).
Here we find that the high-energy even mode ($\sim 18$ meV) is isotropic in
spin space, but the low-energy odd modes consist of a $c-$axis polarized mode
around 9 meV along with another partially overlapped in-plane mode around 12
meV. We argue that such spin anisotropy is induced by the spin-orbit coupling
in the spin-vortex-type fluctuations of this unique compound. The spin
anisotropy is strongly affected by the superconductivity, where it is weak
below 6 meV in the normal state and then transferred to higher energy and
further enhanced in the odd mode of spin resonance below $T_c$.
