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.