Three apparently independent lines of investigation suggest that the four-component Dirac spinor contains a deeper two-state structure than is usually recognised. Peter Woit [1] observed that a Standard Model lepton generation can be identified with components of the spinor representation associated with a natural U(2) structure. Robert A. Wilson has argued [2] that the Dirac algebra possesses an intrinsic doubled geometry associated with so(4,2). Independently, a vacuum-polarisation model of particle structure suggests that the primitive physical states are not the conventional electron chiralities but instead the pair (ν_L, e_R) [3]. We compare these three approaches and show that, although they arise from very different motivations, they all focus attention on a common two-state structure underlying the Dirac equation. This suggests that the familiar decomposition of the Dirac spinor into left- and right-handed components may not be the unique physical interpretation of the underlying mathematics.