  We describe a novel duality symmetry of Phi(4)-theory defined on
noncommutative Euclidean space and with noncommuting momentum coordinates. This
duality acts on the fields by Fourier transformation and scaling. It is an
extension, to interactions defined with a star-product, of that which arises in
quantum field theories of non-interacting scalar particles coupled to a
constant background electromagnetic field. The dual models are in general of
the same original form but with transformed coupling parameters, while in
certain special cases all parameters are essentially unchanged. Using a
particular regualarization we show, to all orders of perturbation theory, that
that this duality also persists at the quantum level. We also point out various
other properties of this class of noncommutative field theories.
