  For w-legged antiferromagnetic spin-1/2 Heisenberg ladders, a long-range
spin-pairing order can be identified which enables the separation of the space
spanned by finite-range (covalent) valence-bond configurations into w+1
subspaces. Since every subspace has an equivalent counter subspace connected by
translational symmetry, twofold degeneracy, breaking traslational symmetry is
found except for the subspace where the ground state of w=even belongs to. In
terms of energy ordering, (non)degeneracy and the discontinuities introduced in
the long-range spin-pairing order by topological spin defects, the differences
between even and odd ladders are explained in a general and systematic way.
