We report high-field magnetization, high-frequency electron spin resonance (ESR), and 77 Se nuclear magnetic resonance (NMR) measurements on the linear spin tetramer system CuSeO 3 , consisting of strongly interacting Cu(1) dimers and weakly coupled Cu(2) spins. The magnetization exhibits anisotropic half-step magnetization plateaux at µ 0 H = 45 T, depending on a crystallographic orientation. A temperature dependence of the ESR linewidth ∆H pp in a paramagnetic phase points towards the significance of anisotropic exchange interactions. Below T N = 9 − 10 K long-range magnetic order is evidenced by the observation of a critical divergence of both ∆H pp (T ) and the nuclear spin-lattice relaxation rate 1/T 1 . In addition, we identify a magnetic anomaly at T * = 6.0(5) K below T N , which is caused by a spin reorientation. The nuclear spin-spin relaxation rate 1/T 2 unveils the development of site-specific spin correlations. The intriguing magnetism of CuSeO 3 is discussed in terms of the energy hierarchy of Cu(1) and Cu(2) spins in concert with additional intertetramer interactions.