  We study a novel Fulde-Ferrell equal-spin pairing state with opposite center
of mass Cooper pair momentum for each spin polarization. This state respects
time-reversal symmetry and is dubbed a Kramers Fulde-Ferrell (KFF) state. It
can be realized in a one-dimensional system with spin-orbit coupling and
nearest neighbor attraction. We find that the KFF state supports nonreciprocal
spin transport for both bulk superconductor and Josephson junctions when
inversion symmetry is broken. In addition to the spin Josephson diode effect,
the charge transport is controlled by intriguing dynamics of bound states whose
transitions can be manipulated by the length of the KFF superconductor. We
discuss the relevance of the effective theory to the novel physics observed in
monolayer Fe-based superconductor along line defects and domain walls, and the
potential for using spins to make dissipationless superconducting spintronics.
