Abstract: We consider pure SU(N) gauge theories defined on an orbifold lattice, analogous to the S^1/Z_2 gauge theory orbifolds of the continuum, which according to the perturbative analysis do not have a Higgs phase. Non-perturbatively the conclusion for N even is the opposite, namely that spontaneous symmetry breaking does take place and some of the gauge bosons become massive. Mathematically this is a result of the spontaneous breaking of the generalized Weyl group of the 5d bulk gauge symmetry and physically it is seen as a kind of relativistic, bosonic superconductivity.