  The entanglement spectrum provides crucial information about correlated
quantum systems. We show that the study of the block-like nature of the reduced
density matrix in number sectors and the partition dependence of the spectrum
in finite systems leads to interesting unexpected insights, which we illustrate
for the case of a 1D extended Hubbard model. We show that block symmetry
provides an intuitive understanding of the spectral double degeneracy of the
Haldane-insulator, revealing as well partial double degeneracy for the
Mott-insulator. Moreover, surprisingly, the partition dependence of the
spectral degeneracy in the Haldane- and Mott-insulator is directly linked to
the, in principle unrelated, density-density correlations, and presents an
intriguing periodic behavior in superfluid and supersolid phases.
