Understanding the structure of the wavefunction is essential for depicting the surface states of a topological insulator. Owing to the inherent strong spin–orbit coupling, the conventional hand-waving picture of the Dirac surface state with a single chiral spin texture is incomplete, as this ignores the orbital components of the Dirac wavefunction and their coupling to the spin textures. Here, by combining orbital-selective angle-resolved photoemission experiments and first-principles calculations, we deconvolve the in-plane and out-of-plane p-orbital components of the Dirac wavefunction. The in-plane orbital wavefunction is asymmetric relative to the Dirac point. It is predominantly tangential (radial) to the k-space constant energy surfaces above (below) the Dirac point. This orbital texture switch occurs exactly at the Dirac point, and therefore should be intrinsic to the topological physics. Our results imply that the Dirac wavefunction has a spin–orbital texture—a superposition of orbital wavefunctions coupled with the corresponding spin textures.