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The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediates collinear magnetic interactions via the conduction electrons of a non-magnetic spacer, resulting in a ferro- or antiferromagnetic magnetization in magnetic multilayers. The resulting spin-polarized charge transport effects have found numerous applications. Recently it has been discovered that heavy non-magnetic spacers are able to mediate an indirect magnetic coupling that is non-collinear and chiral. This Dzyaloshinskii-Moriya-enhanced RKKY (DME-RKKY) interaction causes the emergence of a variety of interesting magnetic structures, such as skyrmions and spin spirals. Applications using these magnetic quasi-particles require a thorough understanding and fine-tuning of the balance between the Dzyaloshinskii-Moriya interaction and other magnetic interactions, e.g., the exchange interaction and magnetic anisotropy contributions. Here, we show by spin-polarized scanning tunneling microscopy that the spin structure of manganese oxide chains on Ir(001) can reproducibly be switched from chiral to collinear antiferromagnetic interchain interactions by increasing the oxidation state of MnO$_2$ while the reverse process can be induced by thermal reduction. The underlying structural change is revealed by low-energy electron diffraction intensity data (LEED-IV) analysis. Density functional theory calculations suggest that the magnetic transition may be caused by a significant increase of the Heisenberg exchange upon oxidation.