  The qualitative difference in low-energy properties of spin $S$ quantum
antiferromagnetic chains with integer $S$ and half-odd-integer $S$ discovered
by Haldane can be intuitively understood in terms of the valence-bond picture
proposed by Affleck, Kennedy, Lieb, and Tasaki. Here we develop a similarly
intuitive diagrammatic explanation of the qualitative difference between chains
with odd $S$ and even $S$, which is at the heart of the theory of
symmetry-protected topological (SPT) phases.
  More precisely, we define one-parameter families of states, which we call the
asymmetric valence-bond solid (VBS) states, that continuously interpolate
between the Affleck-Kennedy-Lieb-Tasaki (AKLT) state and the trivial zero state
in quantum spin chains with $S=1$ and 2. The asymmetric VBS state is obtained
by systematically modifying the AKLT state. It always has exponentially
decaying truncated correlation functions and is a unique gapped ground state of
a short-ranged Hamiltonian. We also observe that the asymmetric VBS state
possesses the time-reversal, the $\mathbb{Z}_2\times\mathbb{Z}_2$, and the
bond-centered inversion symmetries for $S=2$, but not for $S=1$. This is
consistent with the known fact that the AKLT model belongs to the trivial SPT
phase if $S=2$ and to a nontrivial SPT phase if $S=1$. Although such
interpolating families of disordered states were already known, our
construction is unified and is based on a simple physical picture. It also
extends to spin chains with general integer $S$ and provides us with an
intuitive explanation of the essential difference between models with odd and
even spins.
