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Of all the parent compounds of iron-based high-temperature superconductors, EuFe₂As₂ exhibits by far the largest magnetostructural coupling due to the strong interaction between localized Eu 4f moments and the Fe spin density wave order. However, the formation of structural/antiferromagnetic domains couples the Eu antiferromagnetic order to a series of complex field detwinning processes, which impedes a simple magnetometry measurement from extracting the critical fields of the metamagnetic transition. Here we make a measurement of these critical fields by combining x-ray magnetic circular dichroism with in-situ tunable uniaxial stress and applied magnetic field. The combination of two tuning knobs allows us to separate the stress-detwinning of structural domains from the field reorientation of Eu moments. Intriguingly, the spin flip and saturation field values reveal that in addition to the effective single ion anisotropy from a Eu-Fe biquadratic interaction, the Eu interplanar exchange is also strongly anisotropic. This anisotropic exchange term is likely a consequence of the strong anisotropy in the magnetically ordered Fe layer, revealing the non-trivial interplay between Eu-Fe magnetism in this system.