Contacting ferromagnetic films with normal metals changes how magnetic textures respond to electric currents, enabling surprisingly fast domain wall motions and spin texture-dependent propagation direction. These effects are attributed to domain wall chirality induced by the Dzyaloshinskii-Moriya interaction at interfaces, which suggests rich possibilities to influence domain wall dynamics if the Dzyaloshinskii-Moriya interaction can be adjusted. Chiral magnetism was seen in several film structures on appropriately chosen substrates where interfacial spin-orbit-coupling effects are strong. Here we use real-space imaging to visualize chiral domain walls in cobalt/nickel multilayers in contact with platinum and iridium. We show that the Dzyaloshinskii-Moriya interaction can be adjusted to stabilize either left-handed or right-handed Néel walls, or non-chiral Bloch walls by adjusting an interfacial spacer layer between the multilayers and the substrate. Our findings introduce domain wall chirality as a new degree of freedom, which may open up new opportunities for spintronics device designs.