$\mathrm{Cu}{(\mathrm{pz})}_{2}{({\mathrm{ClO}}_{4})}_{2}$ (with pz denoting pyrazine, ${\mathrm{C}}_{4}{\mathrm{H}}_{4}{\mathrm{N}}_{2})$ is a two-dimensional spin-1/2 square-lattice antiferromagnet with ${T}_{\mathrm{N}}=4.24$ K. Due to a persisting focus on the low-temperature magnetic properties, its room-temperature structural and physical properties caught no attention up to now. Here we report a study of the structural features of $\mathrm{Cu}{(\mathrm{pz})}_{2}{({\mathrm{ClO}}_{4})}_{2}$ in the paramagnetic phase, up to 330 K. By employing magnetization, specific heat, $^{35}\mathrm{Cl}$ nuclear magnetic resonance, and neutron diffraction measurements, we provide evidence of a second-order phase transition at ${T}^{★}=294$ K, not reported before. The absence of a magnetic ordering across ${T}^{★}$ in the magnetization data, yet the presence of a sizable anomaly in the specific heat, suggests a structural order-to-disorder-type transition. NMR and neutron-diffraction data corroborate our conjecture by revealing subtle angular distortions of the pyrazine rings and of ${\mathrm{ClO}}_{4}^{\ensuremath{-}}$ counteranion tetrahedra, shown to adopt a configuration of higher symmetry above the transition temperature.