translation operator also involves a gauge transformation, leaving a preferred section and inducing SSB. Choosing a specific twist means to choose the VEV alignment.

Wilson lines and the Poincaré group
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We can generalize any of the transformations generated by the extra dimensional Poincaré group, not just translations. To generalize the rotation operator we need at least 2 extra dimensions.

Wilson lines are generalizations of the extra dimensional translation operators which generate an effective VEV and spontaneous symmetry breaking. Orbifold identifications are related to generalizations of the extra dimensional parity operators $\mathcal{P}_m$ and extra dimensional rotations which are defined by the Lorentz generators $M_{mn}$. These break the symmetry at low energies by only leaving zero modes to the fields that commute with the chosen gauge transformation. The orbifold identification must be a discrete operation from the group $SO(M-4)\times \mathbb{Z}_2^{M-4}$, leaving the 4d Poincaré group unbroken.

These generalizations can be applied simultaneously to the extra dimensional part of the Poincaré group $SO(M,1)\ltimes T^M$ when complying with its algebra $$\begin{split}[P_m, P_n] &= 0,\\
\frac{ 1 }{ i }~[M_{mn}, P_r] &= \eta_{mr} P_n - \eta_{nr} P_m,\\
\frac{ 1 }{ i }~[M_{mn}, M_{rs}] &= \eta_{mr} M_{ns} - \eta_{ms} M_{nr} - \eta_{nr} M_{ms} + \eta_{ns} 