In the basic interacting-boson model (sd IBA-1), as well as in other one-fluid collective models of nuclear structure, the magnetic dipole operator in leading order is proportional to the angular momentum and cannot produce transitions. Phenomenological second-order terms are therefore needed. The origin of these terms in IBA-1 is traced back to the proton-neutron two-fluid model IBA-2 in two ways. First, an IBA-2 second-order M1 operator is proposed as a mapping of the linear operator to the space of pure F spin and the second-order operator is projected onto IBA-1 space. Second, the linear operator of IBA-2 is mapped directly to the quadratic operator of IBA-1. Explicit calculations are carried out in a pure and a perturbed U(5) limit.