  We investigate the symmetry algebras of integrable spin Calogero systems
constructed from Dunkl operators associated to finite Coxeter groups. Based on
two explicit examples, we show that the common view of associating one symmetry
algebra to a given Coxeter group $W$ is wrong. More precisely, the symmetry
algebra heavily depends on the representation of $W$ on the spins. We prove
this by identifying two different symmetry algebras for a $B_L$ spin Calogero
model and three for $G_2$ spin Calogero model. They are all related to the
half-loop algebra and its twisted versions. Some of the result are extended to
any finite Coxeter group.
