  Calculations in the sd and pf shells reported some time ago by Satu\l a\etal\
[Phys.~Lett.~B~407, 103 (1997)] are redone for an extended analysis of the
results. As in the original work, we do calculations for one mass number in
each shell and consider in each case the sequence of lowest energies for
isospins 0, 2, and 4, briefly the symmetry spectrum. Following further the
original work we study how this spectrum changes when parts of the two-nucleon
interaction are turned off. The variation of its width is explored in detail. A
differential combination $\epsilon_\text{W}$ of the three energies was taken in
the original work as a measure of the so-called Wigner term in semi-empirical
mass formulas, and it was found to decrease drastically when the two-nucleon
interaction in the channel of zero isospin is turned off. Our analysis shows
that the width of the symmetry spectrum experiences an equally drastic
decrease, which can be explained qualitatively in terms of schematic
approximations. We therefore suggest that the decrease of $\epsilon_\text{W}$
be seen mainly as a side effect of a narrowing of the symmetry spectrum rather
than an independent manifestation of the two-nucleon interaction in the channel
of zero isospin.
