By applying Darboux-Crum transformations to a Lax pair formulation of the Korteweg-de Vries (KdV) equation, we construct new sets of multisoliton solutions to it as well as to the modified Korteweg-de Vries (mKdV) equation. The obtained solutions exhibit a chiral asymmetry in the propagation of different types of defects in crystalline backgrounds. We show that the KdV solitons of pulse and compression modulation types-which support bound states in, respectively, semi-infinite and finite forbidden bands in the spectrum of the perturbed quantum one-gap Lame system-propagate in opposite directions with respect to the asymptotically periodic background. A similar but more complicated picture also appears for multi-kink-antikink mKdV solitons that propagate with a privileged direction over the topologically trivial or topologically nontrivial crystalline background depending on the position of the energy levels of the trapped bound states in the spectral gaps of the associated Dirac system. An exotic N = 4 nonlinear supersymmetric structure incorporating Lax-Novikov integrals of a pair of perturbed Lame systems is shown to underlie the Miura-Darboux-Crum construction. It unifies the KdV and mKdV solutions, detects the defects and distinguishes their types, and identifies the types of crystalline backgrounds.