In this paper it is shown that the Banach space of continuous, R^2- or C-valued functions on a simply connected either 2-dimensional real or 1-dimensional complex compact region can be decomposed into the topological direct sum of two subspaces, a subspace of integrable (and conformal) functions, and another one of unintegrable (and anti-conformal) functions. It is shown that complex integrability is equivalent to complex analyticity. This can be extended to real functions. The existence of a conjugation on that Banach space will be proven, which maps unintegrable functions onto integrable functions.The boundary of a 2-dimensional simply connected compact region is defined by a Jordan curve, from which it is known to topologically divide the domain into two disconnected regions. The choice of which of the two regions is to be the inside, defines the orientation.The conjugation above will be seen to be the inversion of orientation.Analyticity, integrability, and orientation on R^2 (or C) therefore are intimately related.