  The Jordan-Wigner transformation is frequently utilised to rewrite quantum
spin chains in terms of fermionic operators. When the resulting Hamiltonian is
bilinear in these fermions, i.e. the fermions are free, the exact spectrum
follows from the eigenvalues of a matrix whose size grows only linearly with
the volume of the system. However, several Hamiltonians that do not admit a
Jordan-Wigner transformation to fermion bilinears still have the same type of
free-fermion spectra. The spectra of such ``free fermions in disguise" models
can be found exactly by an intricate but explicit construction of the raising
and lowering operators. We generalise the methods further to find a family of
such spin chains. We compute the exact spectrum, and generalise an elegant
graph-theory construction. We also explain how this family admits an N=2
lattice supersymmetry.
