  FeSe is arguably the simplest, yet the most enigmatic, iron-based
superconductor. Its nematic but non-magnetic ground state is unprecedented in
this class of materials and stands out as a current puzzle. Here, our NMR
measurements in the nematic state of mechanically detwinned FeSe reveal that
both the Knight shift and the spin-lattice relaxation rate 1/T_1 possess an
in-plane anisotropy opposite to that of the iron pnictides LaFeAsO and
BaFe2As2. Using a microscopic electron model that includes spin-orbit coupling,
our calculations show that an opposite quasiparticle weight ratio between the
d_xz and d_yz orbitals leads to an opposite anisotropy of the orbital magnetic
susceptibility, which explains our Knight shift results. We attribute this
property to a different nature of nematic order in the two compounds,
predominantly bond-type in FeSe and onsite ferro-orbital in pnictides. The T_1
anisotropy is found to be inconsistent with existing neutron scattering data in
FeSe, showing that the spin fluctuation spectrum reveals surprises at low
energy, possibly from fluctuations that do not break C_4 symmetry. Therefore,
our results reveal that important information is hidden in these anisotropies
and they place stringent constraints on the low-energy spin correlations as
well as on the nature of nematicity in FeSe.
