  We introduce a new class of open, translationally invariant spin chains with
long-range interactions depending on both spin permutation and (polarized) spin
reversal operators, which includes the Haldane-Shastry chain as a particular
degenerate case. The new class is characterized by the fact that the
Hamiltonian is invariant under "twisted" translations, combining an ordinary
translation with a spin flip at one end of the chain. It includes a remarkable
model with elliptic spin-spin interactions, smoothly interpolating between the
XXX Heisenberg model with anti-periodic boundary conditions and a new open
chain with sites uniformly spaced on a half-circle and interactions inversely
proportional to the square of the distance between the spins. We are able to
compute in closed form the partition function of the latter chain, thereby
obtaining a complete description of its spectrum in terms of a pair of
independent su(1|1) and ${\rm su}(m/2)$ motifs when the number $m$ of internal
degrees of freedom is even. This implies that the even $m$ model is invariant
under the direct sum of the Yangians $Y$(gl(1|1)) and $Y$(gl$(0|m/2)$). We also
analyze several statistical properties of the new chain's spectrum. In
particular, we show that it is highly degenerate, which strongly suggests the
existence of an underlying (twisted) Yangian symmetry also for odd $m$.
