<p>This paper presents a characterization of the Method of Fundamental Solutions for solving 2D Dirichlet problems from a fundamental perspective. By contrasting the inherent characteristics of the conventional and invariant schemes, we show that in particular cases these two schemes are partially equivalent with regard to their linear systems, where a decomposition by using orthogonalization of the problem into two systems highlights the invariant subspaces of the coefficient matrix. Moreover, we explore their inherent reciprocity properties with a particular focus on their values of sources and boundary values. A key contribution of this work is the proofs of reciprocity preservation in both the conventional and invariant schemes, and providing examples and remarks on them. We demonstrate that this property holds between inner and outer problems, regardless of the domain’s shape complexity. As an application of the reciprocity property, we demonstrate that an approximate solution can be obtained by solving underdetermined systems, even in cases where some collocation points are missing. The numerical experiments suggest that, the reciprocity-based method may even surpass the desirable effects of regularization—namely, suppressing error while keeping the residual small.</p>