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.
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