One Twist, Two Views — the connection between the tensor-invariant system and the Möbius–Klein transform

Two independently built frameworks – the unified tensor-invariant system of vector analysis (grad/jad/dev operators, dimensional ladder) and the spectral theory of non-orientable surfaces (Möbius and Klein transforms, sector decomposition) – rest on a single common actor: orientation. In the tensor system orientation hides inside the formulas (the sign flip of the epsilon -sorter under reflection: the “pseudotensor birthmark”); in the Möbius–Klein framework orientation is built into the space itself (the space carries a built-in “mirror”). This paper shows step by step that the two frameworks are two views of the same twist – local-differential and global-spectral –, writes down the precise dictionary between them, and derives the result at their meeting point, corroborating it numerically: on the Klein bottle the curl switches sector, and the stream function of the Helmholtz decomposition lives in the twisted sector. The same rule then runs through quantum mechanics and spinors: the two sectors are two theories of a one-parameter quantization family that the geometry singles out, the twisted one carrying a zero-point gap; the Pin class of a spinor fixes its frequency lattice, and chirality switches sector, so within a sector there is no net chirality. Curl, angular momentum and chirality turn out to be three instances of one selection theorem — parity against block structure, the observable part being the even component — and one dimension up the same criterion makes the curl split component-wise instead of switching. As a quantitative consequence, the Casimir energy of a vector field on the Klein bottle is the sum of its two sectors rather than twice the scalar value, and the four Pin structures give vacuum energies that differ even in sign. We then discuss what all this means in physics – from the mirror behaviour of the magnetic field to fermionic boundary conditions – and in engineering practice; the cosmological outlook states the E–B sign-signature conjecture and the twisted-compactification program, and takes up passage and time, from the Lorentz boost to the Matsubara temperature. One row of the dictionary is sharpened: rot and jad are the same slice in different readings – the tensor form of jad carries no epsilon and is single-valued on the Klein bottle, while rot as the dual reading switches sector, so on a non-orientable space it is better to compute in jad. From this a decidable criterion follows for the whole ladder: a rung carries over unchanged if and only if its selector contains an even number of epsilon 's. Finally, a consequence for materials physics on the third-order rung: on the orientation-reversing loop the component of the Burgers vector along the mirrored direction changes sign, so the vector is not a global constant but a sector-dependent quantity.