  In the layered cuprate perovskites, the occurence of high-temperature
superconductivity seems deeply related to the unusual nature of the hole
excitations. The limiting case of a very small number of holes diffusing in the
antiferromagnetic (AF) background may provide important insights into this
problem. We have investigated the transport properties in a series of crystals
of $\rm YBa_2Cu_3O_y$, and found that the temperature dependences of the Hall
coefficient $R_H$ and thermopower $S$ change abruptly as soon as the AF phase
boundary is crossed. In the AF state at low temperatures $T$, both $R_H$ and
$S$ are unexpectedly suppressed to nearly zero over a broad interval of $T$. We
argue that this suppression arises from near-exact symmetry in the
particle-hole currents. From the trends in $R_H$ and $S$, we infer that the
symmetry is increasingly robust as the hole density $x$ becomes very small
($x\simeq 0.01$). We discuss implications for electronic properties both within
the AF state and outside.
