Combining first-principles calculations and spin-and angle-resolved photoemission spectroscopy measurements, we identify the helical spin textures for three different Dirac cone states in the interfaced systems of a two-dimensional (2D) topological insulator (TI) of a Bi(111) bilayer and a three-dimensional (3D) TI B or Bi2Te3. The spin texture is found to be the same for the intrinsic Dirac cone of Bi2Te3 or Bi2Te3 surface states, the extrinsic Dirac cone of Bi bilayer induced by the Rashba effect, and the hybridized Dirac cone between the former two states. Further orbit-and atom-resolved analysis shows that s and p(z) orbits have a clockwise (counterclockwise) spin rotation tangent to the iso-energy contour of the upper (lower) Dirac cone, while p(x) and p(y) orbits have radial spin components. The Dirac cone states may reside on different atomic layers, but have the same spin texture. Our results suggest that the unique spin texture of Dirac cone states is a signature property of spin-orbit coupling, independent of topology.