  A set of interacting particles are coupled to a phenomenological core
described using the generalized coherent state model. Among the particle-core
states a finite set which have the property that the angular momenta carried by
the proton and neutron quadrupole bosons and the particles, separately, are
mutually orthogonal are identified. The magnetic properties of such states are
studied. All terms of the model Hamiltonian exhibit chiral symmetry except the
spin-spin interaction. There are four bands of the type with
two-quasiparticle-core dipole states, exhibiting properties which are specific
for magnetic twin bands. An application is presented, for the isotopes $^{188,
190}$Os.
