  We argue that classical pyrochlore Heisenberg antiferromagnets with small
further-neighbor couplings can order in a state where pairs of sublattices form
antiparallel spirals. The spiral ordering wave vectors of the two pairs are in
general different from each other, and are constrained by which sublattices are
being paired. This sublattice pairing state generally breaks inversion and most
rotation symmetries. Its existence depends on the antiferromagnetic
nearest-neighbor coupling which favors the spins on each tetrahedron to sum to
zero. To substantiate our argument, we extend the nematic bond theory; a
diagrammatic large-$N_s$ method, to non-Bravais lattices, and we demonstrate
that the predicted state is indeed realized at low temperatures in a large
region of exchange coupling space. We also carry out a spin wave calculation
which suggests that the sublattice pairing state is coplanar.
