  We refine the recently introduced notion of eclectic spin chains by including
a maximal number of deformation parameters. These models are integrable,
nearest-neighbor n-state spin chains with exceedingly simple non-hermitian
Hamiltonians. They turn out to be non-diagonalizable in the multiparticle
sector (n>2), where their "spectrum" consists of an intricate collection of
Jordan blocks of arbitrary size and multiplicity. We show how and why the
quantum inverse scattering method, sought to be universally applicable to
integrable nearest-neighbor spin chains, essentially fails to reproduce the
details of this spectrum. We then provide, for n=3, detailed evidence by a
variety of analytical and numerical techniques that the spectrum is not
"random", but instead shows surprisingly subtle and regular patterns that
moreover exhibit universality for generic deformation parameters. We also
introduce a new model, the hypereclectic spin chain, where all parameters are
zero except for one. Despite the extreme simplicity of its Hamiltonian, it
still seems to reproduce the above "generic" spectra as a subset of an even
more intricate overall spectrum. Our models are inspired by parts of the
one-loop dilatation operator of a strongly twisted, double-scaled deformation
of N=4 Super Yang-Mills Theory.
