  We study quantum many-body systems in the presence of an exotic antiunitary
translation or inversion symmetry involving time reversal. Based on a
symmetry-twisting method and spectrum robustness,we propose that a half-integer
spin chain which respects any of these two antiunitary crystalline symmetries
in addition to the discrete $\mathbb{Z}_2\times\mathbb{Z}_2$ global
spin-rotation symmetry must either be gapless or possess degenerate ground
states. This explains the gaplessness of a class of chiral spin models not
indicated by the Lieb-Schultz-Mattis (LSM) theorem and its known extensions.
Moreover, we present symmetry classes with minimal sets of generators that give
nontrivial LSM-type constraints, argued by the bulk-boundary correspondence in
2d symmetry-protected topological phases as well as lattice homotopy. Our
results for detecting the ingappability of 1d quantum magnets from the
interplay between spin-rotation symmetries and magnetic space groups are
applicable to systems with a broader class of spin interactions, including the
Dzyaloshinskii-Moriya and triple-product interactions.
