We report on the controlled switching of domain-wall (DW) magnetization in aligned stripe-domain structures, stabilized in [Co(0.44 nm)/Pt(0.7 nm)]X(X=48
, 100, 150) multilayers with perpendicular magnetic anisotropy. The switching process, induced by an external magnetic field, is monitored by measuring the evolution of the in-plane magnetization. We show that the remanent in-plane magnetization originates from the polarization of the Bloch-type DWs. With micromagnetic simulations, we reveal that the reversal of the DW polarization is the result of the emergence and collapse of horizontal Bloch lines within the DWs at particular strengths of the external magnetic field, applied opposite to the DW polarization. Our findings are relevant for DW-based magnonics and bubble-skyrmion applications in magnetic multilayers.