  Cu(pz)$_2$(ClO$_4$)$_2$ (with pz denoting pyrazine C$_4$H$_4$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
Cu(pz)$_2$(ClO$_4$)$_2$ in the paramagnetic phase, up to 330 K. By employing
magnetization, specific heat, $^{35}$Cl nuclear magnetic resonance, and neutron
diffraction measurements, we provide evidence of a second-order phase
transition at $T^{\star}$ = 294 K, not reported before. The absence of a
magnetic ordering across $T^{\star}$ in the magnetization data, yet the
presence of a sizable anomaly in the specific heat, suggest 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 ClO$^-_4$ counteranion tetrahedra, shown to adopt a configuration of
higher symmetry above the transition temperature.
