  In this work, we exhaust all the spin-space symmetries, which fully
characterize collinear, non-collinear, commensurate, and incommensurate spiral
magnetism, and investigate enriched features of electronic bands that respect
these symmetries. We achieve this by systematically classifying the so-called
spin space groups (SSGs) - joint symmetry groups of spatial and spin operations
that leave the magnetic structure unchanged. Generally speaking, they are
accurate (approximate) symmetries in systems where spin-orbit coupling (SOC) is
negligible (finite but weaker than the interested energy scale); but we also
show that specific SSGs could remain valid even in the presence of a strong
SOC. By representing the SSGs as O($N$) representations, we - for the first
time - obtain the complete classifications of 1421, 9542, and 56512 distinct
SSGs for collinear ($N=1$), coplanar ($N=2$), and non-coplanar ($N=3$)
magnetism, respectively. SSG not only fully characterizes the symmetry of spin
d.o.f., but also gives rise to exotic electronic states, which, in general,
form projective representations of magnetic space groups (MSGs). Surprisingly,
electronic bands in SSGs exhibit features never seen in MSGs, such as
nonsymmorphic SSG Brillouin zone (BZ), where SSG operations behave as glide or
screw when act on momentum and unconventional spin-momentum locking, which is
completely determined by SSG, independent of Hamiltonian details. To apply our
theory, we identify the SSG for each of the 1604 published magnetic structures
in the MAGNDATA database on the Bilbao Crystallographic Server. Material
examples exhibiting aforementioned novel features are discussed with emphasis.
We also investigate new types of SSG-protected topological electronic states
that are unprecedented in MSGs.
