This paper develops matroidal analogues of classical results on matchings in abelian groups. By embedding matroid ground sets in an abelian group, we introduce base matchings between matroid bases, recover the group-theoretic setting in the uniform matroid case, and derive structural and combinatorial criteria for their existence. Our main focus is on paving matroids. We prove self-matchability for paving matroids, extend asymmetric matchability results using the hyperplane-nullity parameter, and show that stressed hyperplanes provide a natural route to matchability through relaxation.