  We develop the basics of twistor theory in de Sitter space, up to the Penrose
transform for free massless fields. We treat de Sitter space as fundamental, as
one does for Minkowski space in conventional introductions to twistor theory.
This involves viewing twistors as spinors of the de Sitter group SO(4,1). When
attached to a spacetime point, such a twistor can be reinterpreted as a local
SO(3,1) Dirac spinor. Our approach highlights the antipodal map in de Sitter
space, which gives rise to doublings in the standard relations between twistors
and spacetime. In particular, one can generate a field with both handedness
signs from a single twistor function. Such fields naturally live on
antipodally-identified de Sitter space dS_4/Z_2, which has been put forward as
the ideal laboratory for quantum gravity with positive cosmological constant.
