Methods to generate spin-polarised electronic states in non-magnetic solids are strongly desired to enable all-electrical manipulation of electron spins for new quantum devices. This is generally accepted to require breaking global structural inversion symmetry. In contrast, here we present direct evidence from spin- and angle-resolved photoemission spectroscopy for a strong spin polarisation of bulk states in the centrosymmetric transition-metal dichalcogenide WSe$_2$. We show how this arises due to a lack of inversion symmetry in constituent structural units of the bulk crystal where the electronic states are localised, leading to enormous spin splittings up to $sim!0.5$ eV, with a spin texture that is strongly modulated in both real and momentum space. As well as providing the first experimental evidence for a recently-predicted `hidden spin polarisation in inversion-symmetric materials, our study sheds new light on a putative spin-valley coupling in transition-metal dichalcogenides, of key importance for using these compounds in proposed valleytronic devices.
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