  We construct gauge theories in two extra dimensions compactified on the
chiral square, which is a simple compactification that leads to chiral fermions
in four dimensions. Stationarity of the action on the boundary specifies the
boundary conditions for gauge fields. Any six-dimensional gauge field
decomposed in Kaluza-Klein modes includes a tower of heavy spin-1 particles
whose longitudinal polarizations are linear combinations of the
extra-dimensional components, and a tower of heavy spin-0 particles
corresponding to the orthogonal combinations. These linear combinations depend
on the Kaluza-Klein numbers, and are independent of the gauge fixing. If the
gauge symmetry is broken by the vacuum expectation value of a six-dimensional
scalar, at each Kaluza-Klein level three spinless fields in the adjoint
representation mix to provide the longitudinal polarization of the spin-1 mode,
leaving the orthogonal states as two spin-0 particles. We derive the
interactions of the Kaluza-Klein modes for generic gauge theories, laying the
groundwork for the Standard Model in two universal extra dimensions, and more
generally for future model building and phenomenological studies.
