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