Several inequivalent data at a Lorentzian spin-foam vertex are commonly called "orientation."We separate face coorientation, tetrahedral handedness, time orientation of a tetrahedral normal, four-simplex branch, causal wedge data, and global time orientability, and classify proposed reversals by the carrier and category in which they act. Inversion of an integrated SL(2,C) variable is only a Haar-measure reparametrization. By contrast, an SU(2) coherent-state spinor is a square root of its face flux: the standard antilinear spinor conjugation sends the flux to its antipode. On a spacelike quantum tetrahedron the tensor product of these antiunitaries fixes every area, reverses the oriented-volume operator, and squares to +1 on every admissible intertwiner space; hence nonzero-volume eigenstates are paired orthogonally, but no Kramers degeneracy follows. For a timelike tetrahedron, the intrinsic algebra is Cl(1, 2). The sign of a face bivector square gives the elliptic—parabolic—hyperbolic orbit trichotomy. The two sheets associated with a spacelike face carry the discrete series D±k of connected SU(1, 1): there is no bounded linear intertwiner and no continuous invariant sesquilinear pairing between them, although a unique bounded invariant bilinear pairing and antiunitary or disconnected twisted exchanges exist. Finally, in every intrinsic odd-dimensional Clifford space, total inversion is the twisted-adjoint action of the odd pseudoscalar and cannot be implemented by an even intrinsic versor. Orientation protection is therefore relative to a specified carrier, symmetry group, and morphism category; a conservation law requires an explicit multi-vertex gluing rule.