  The pseudogap phenomenon in cuprates is the most mysterious puzzle in the
research of high-temperature superconductivity. In particular, whether the
pseudogap is associated with a crossover or phase transition has been a
long-standing controversial issue. The tetragonal cuprate
HgBa$_2$CuO$_{4+\delta}$, with only one CuO$_2$ layer per primitive cell, is an
ideal system to tackle this puzzle. Here, we measure the anisotropy of magnetic
susceptibility within the CuO$_2$ plane with exceptionally high-precision
magnetic torque experiments. Our key finding is that a distinct two-fold
in-plane anisotropy sets in below the pseudogap temperature $T^*$, which
provides thermodynamic evidence for a nematic phase transition with broken
four-fold symmetry. Most surprisingly, the nematic director orients along the
diagonal direction of the CuO$_2$ square lattice, in sharp contrast to the bond
nematicity reported in various iron-based superconductors and double-layer
YBa$_2$Cu$_3$O$_{6+\delta}$, where the anisotropy axis is along the Fe-Fe and
Cu-O-Cu directions, respectively. Another remarkable feature is that the
enhancement of the diagonal nematicity with decreasing temperature is
suppressed around the temperature at which short-range charge-density-wave
(CDW) formation occurs. This is in stark contrast to
YBa$_2$Cu$_3$O$_{6+\delta}$, where the bond nematicity is not influenced by the
CDW. Our result suggests a competing relationship between diagonal nematic and
CDW order in HgBa$_2$CuO$_{4+\delta}$.
