The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with (Formula presented.)-independent complex bounded measurable coefficients ((Formula presented.) being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in (Formula presented.), subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in (Formula presented.). Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with (Formula presented.)-independent real (possibly non-symmetric) coefficients there exists a (Formula presented.) such that the Regularity problem is well-posed in (Formula presented.).