  The Rashba effect is one of the most striking manifestations of spin-orbit
coupling in solids, and provides a cornerstone for the burgeoning field of
semiconductor spintronics. It is typically assumed to manifest as a
momentum-dependent splitting of a single initially spin-degenerate band into
two branches with opposite spin polarisation. Here, combining
polarisation-dependent and resonant angle-resolved photoemission measurements
with density-functional theory calculations, we show that the two "spin-split"
branches of the model giant Rashba system BiTeI additionally develop disparate
orbital textures, each of which is coupled to a distinct spin configuration.
This necessitates a re-interpretation of spin splitting in Rashba-like systems,
and opens new possibilities for controlling spin polarisation through the
orbital sector.
