  This is the second in a pair of articles that classify the configuration
space and kinematic symmetry groups for $N$ identical particles in
one-dimensional traps experiencing Galilean-invariant two-body interactions.
These symmetries explain degeneracies in the few-body spectrum and demonstrate
how tuning the trap shape and the particle interactions can manipulate these
degeneracies. The additional symmetries that emerge in the non-interacting
limit and in the unitary limit of an infinitely strong contact interaction are
sufficient to algebraically solve for the spectrum and degeneracy in terms of
the one-particle observables. Symmetry also determines the degree to which the
algebraic expressions for energy level shifts by weak interactions or
nearly-unitary interactions are universal, i.e.\ independent of trap shape and
details of the interaction. Identical fermions and bosons with and without spin
are considered. This article analyzes the symmetries of $N$ particles in
asymmetric, symmetric, and harmonic traps; the prequel article treats the one,
two and three particle cases.
