  We deal with the problem of identifying a background structure and its
perturbation in tetrad theories of gravity. Starting from a peculiar trivial
principal bundle we define a metric which depends only on the gauge connection.
We find the allowed four-dimensional structure groups; two of them turn out to
be the translation group T_4 and the unitary group U(2). When the curvature
vanishes the metric reduces to its background form which coincides with
Minkowski flat metric for the T_4 case and with the Einstein static universe
metric for the U(2) case. The perturbation has a coordinate independent
definition and allows for the introduction of observables distinguished from
those obtained from the metric alone. Finally, we show that any teleparallel
theory of gravity, and hence general relativity, can be considered as a gauge
theory over the groups introduced.
