  Spin-orbit coupling driven by broken inversion symmetry ($I$) is known to
lead to unusual magnetic response of superconductors, including extremely large
critical fields for spin-singlet superconductors. This unusual response is also
known to appear in materials that have $I$, provided there is local
$I$-breaking: fermions participating in superconductivity reside on crystal
sites that lack $I$. Here we show that this unusual response exists even when
the crystal sites preserve $I$. Indeed, we argue that the symmetry of Kramers
degenerate fermionic pseudospin is more relevant than the local crystal site
symmetry. We examine and classify non-symmorphic materials with momentum space
spin-textures that exhibit an anomalous pseudospin with different symmetry
properties than usual spin-1/2. We find that this anomalous pseudospin does not
depend on the existence of local $I$ breaking crystal sites and it optimizes
the unusual magnetic response traditionally associated with locally
noncentrosymmetric superconductors, dramatically extending the range of
relevant materials. We further show this anomalous pseudospin leads to fully
gapped `nodal' superconductors and provides additional insight into the
breakdown of Blount's theorem for pseudospin triplet superconductors. We apply
our results to UPt$_3$, BiS$_2$-based superconductors, Fe-based
superconductors, and paramagnetic UCoGe.
