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We study the influence of mechanical deformations on the Zeeman and Rashba effects in synthesized transition metal dichalcogenide (TMD) nanotubes and their Janus variants from first principles. In particular, we perform symmetry-adapted density functional theory simulations with spin-orbit coupling to determine the variation in the Zeeman and Rashba splittings with axial and torsional deformations. We find significant splitting in molybdenum and tungsten nanotubes, for which the Zeeman splitting decreases with increase in strain, going to zero for large enough tensile/shear strains, while the Rashba splitting coefficient increases linearly with shear strain, while being zero for all tensile strains, a consequence of the inversion symmetry remaining unbroken. In addition, the Zeeman splitting is relatively unaffected by nanotube diameter, whereas the Rashba coefficient decreases with increase in diameter. Overall, mechanical deformations represent a powerful tool for spintronics in TMD nanotubes as well as their Janus variants., Comment: 14 pages, 4 figures, 1 table