Recent work [hep-th/0504183,hep-th/0508002] indicates an approach to the formulation of diffeomorphism invariant quantum field theories (qfts) on the Groenewold-Moyal (GM) plane. In this approach to the qfts, statistics gets twisted and the S-matrix in the non-gauge qfts becomes independent of the noncommutativity parameter theta^{mu u}. Here we show that the noncommutative algebra has a commutative spacetime algebra as a substructure: the Poincare, diffeomorphism and gauge groups are based on this algebra in the twisted approach as is known already from the earlier work of [hep-th/0510059]. It is natural to base covariant derivatives for gauge and gravity fields as well on this algebra. Such an approach will in particular introduce no additional gauge fields as compared to the commutative case and also enable us to treat any gauge group (and not just U(N)). Then classical gravity and gauge sectors are the same as those for theta^{mu u}=0, but their interactions with matter fields are sensitive to theta^{mu u}. We construct quantum noncommutative gauge theories (for arbitrary gauge groups) by requiring consistency of twisted statistics and gauge invariance. In a subsequent paper (whose results are summarized here), the locality and Lorentz invariance properties of the S-matrices of these theories will be analyzed, and new non-trivial effects coming from noncommutativity will be elaborated. This paper contains further developments of [hep-th/0608138] and a new formulation based on its approach.
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