It is known that constant rank-type constraint qualifications (CQs) imply the Mangasarian-Fromovitz CQ (MFCQ) after a suitable local reparametrization of the feasible set, which involves eliminating redundancies (remove and/or transform inequality constraints into equalities) without changing the feasible set locally. This technique has been mainly used to study the similarities between well-known CQs from the literature. In this paper, we propose a different approach: we define a type of reparametrization that constitutes a CQ by itself. We carry out an in-depth study on such reparametrizations, considering not only those linked to MFCQ but also to any known CQ. We discuss the relationship between these new reparametrizations and the local error bound property. Furthermore, we characterize the set of Lagrange multipliers as the sum of its recession cone with a compact set related to the reparametrizations where MFCQ becomes valid.