Conclusive identification of an altermagnetic state requires going beyond mere symmetry arguments. We illustrate this in a combined computational and experimental study of the rutile-like material CuF₂, which is on the list of predicted altermagnets. Using ab initio and linear spin-wave calculations supplied by magnetization measurements, we show that CuF₂ in its experimental monoclinic structure can be described by a spin-1/2 model of weakly coupled square-lattice layers with the in-plane coupling J₁≃ 115 K and two synergistic antiferromagnetic interplane couplings amounting to 4% and 8% of J₁, respectively. Driven by long-range superexchange, these interlayer couplings are oblique to the square planes, resulting in the unit-cell doubling in the magnetically ordered state, thus effectively suppressing any altermagnetic band splitting. Concurrently, we identify unusually strong Dzyaloshinskii-Moriya interactions, |D|/J₁≃ 0.3, that produce spin canting and, together with order-by-disorder effect, pin the Néel vector to the crystallographic b-axis. Additionally, DM anisotropy promotes magnon band splitting, but these bands remain non-chiral. Our results highlight the importance of relativistic effects even in 3d magnets with altermagnetic symmetries.