  Nematicity has emerged as a key feature of cuprate superconductors, but its
link to other fundamental properties such as superconductivity, charge order
and the pseudogap remains unclear. Here we use measurements of transport
anisotropy in YBa$_2$Cu$_3$O$_y$ to distinguish two types of nematicity. The
first is associated with short-range charge-density-wave modulations in a
doping region near $p = 0.12$. It is detected in the Nernst coefficient, but
not in the resistivity. The second type prevails at lower doping, where there
are spin modulations but no charge modulations. In this case, the onset of
in-plane anisotropy - detected in both the Nernst coefficient and the
resistivity - follows a line in the temperature-doping phase diagram that
tracks the pseudogap energy. We discuss two possible scenarios for the latter
nematicity.
