  We investigate the scaling of the entanglement spectrum and of the R\'enyi
block entropies and determine its universal aspects in the ground state of
critical and noncritical one-dimensional quantum spin models. In all cases, the
scaling exhibits an oscillatory behavior that terminates at the factorization
point and whose frequency is universal. Parity effects in the scaling of the
R\'enyi entropies for gapless models at zero field are thus shown to be a
particular case of such universal behavior. Likewise, the absence of
oscillations for the Ising chain in transverse field is due to the vanishing
value of the factorizing field for this particular model. In general, the
transition occurring at the factorizing field between two different scaling
regimes of the entanglement spectrum corresponds to a quantum transition to the
formation of finite-range, ordered structures of quasi-dimers, quasi-trimers,
and quasi-polymers. This entanglement-driven transition is superimposed to and
independent of the long-range magnetic order in the broken symmetry phase.
Therefore, it conforms to recent generalizations that identify and classify the
quantum phases of matter according to the structure of ground-state
entanglement patterns. We characterize this form of quantum order by a global
order parameter of entanglement defined as the integral, over blocks of all
lengths, of the R\'enyi entropy of infinite order. Equivalently, it can be
defined as the integral of the bipartite single-copy or geometric entanglement.
The global entanglement order parameter remains always finite at fields below
the factorization point and vanishes identically above it.
