  We report measurements of the Seebeck effect in both the $ab$ plane ($S_{\rm
a}$) and along the $c$ axis ($S_{\rm c}$) of the cuprate superconductor
La$_{1.6-x}$Nd$_{0.4}$Sr$_{x}$CuO$_4$ (Nd-LSCO), performed in magnetic fields
large enough to suppress superconductivity down to low temperature. We use the
Seebeck coefficient as a probe of the particle-hole asymmetry of the electronic
structure across the pseudogap critical doping $p^{\star} = 0.23$. Outside the
pseudogap phase, at $p = 0.24 > p^{\star}$, we observe a positive and
essentially isotropic Seebeck coefficient as $T \rightarrow 0$. That $S > 0$ at
$p = 0.24$ is at odds with expectations given the electronic band structure of
Nd-LSCO above $p^{\star}$ and its known electron-like Fermi surface. We can
reconcile this observation by invoking an energy-dependent scattering rate with
a particle-hole asymmetry, possibly rooted in the non-Fermi liquid nature of
cuprates just above $p^{\star}$. Inside the pseudogap phase, for $ p <
p^{\star}$, $S_{\rm a}$ is seen to rise at low temperature as previously
reported, consistent with the drop in carrier density $n$ from $n \simeq 1 + p$
to $n \simeq p$ across $p^{\star}$ as inferred from other transport properties.
In stark contrast, $S_{\rm c}$ at low temperature becomes negative below
$p^{\star}$, a novel signature of the pseudogap phase. The sudden drop in
$S_{\rm c}$ reveals a change in the electronic structure of Nd-LSCO upon
crossing $p^{\star}$. We can exclude a profound change of the scattering across
$p^{\star}$ and conclude that the change in the out-of-plane Seebeck
coefficient originates from a transformation of the Fermi surface.
