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 (jet/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 presents the machine-verified result at their meeting point: on the Klein bottle the curl switches sector, and the stream function of the Helmholtz decomposition lives in the twisted sector. 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: the Burgers vector changes sign on the orientation-reversing loop, so it is not a global constant but a sector-dependent quantity.