  We propose a generalisation of the notion of associated bundles to a
principal bundle constructed via group action cocycles rather than via mere
representations of the structure group. We devise a notion of connection
generalising Ehresmann connection on principal bundles, giving rise to the
appropriate covariant derivative on sections of these twisted associated
bundles (and on twisted tensorial forms). We study the action of the group of
vertical automorphisms on the objects introduced (active gauge
transformations). We also provide the gluing properties of the local
representatives (passive gauge transformations). The latter are generalised
gauge fields: They satisfy the gauge principle of physics, but are of a
different geometric nature than standard Yang-Mills fields. We also examine the
conditions under which this new geometry coexists and mixes with the standard
one. We show that (standard) conformal tractors and Penrose's twistors can be
seen as simple instances of this general picture. We also indicate that the
twisted geometry arises naturally in the definition and study of anomalies in
quantum gauge field theory.
