We are investigating the dynamics of a new Poincaré gauge theory of gravity model, which has cross coupling between the spin-0+ and spin-0- modes. To this end we are considering a very appropriate situation – homogeneous-isotropic cosmologies – which is relatively simple, and yet all the modes have non-trivial dynamics which reveals physically interesting and possibly observable results. More specifically we consider manifestly isotropic Bianchi class A cosmologies. Here the first order equations obtained from an effective Lagrangian are linearized and the normal modes are found. These turn out to control the asymptotic late time cosmological normal modes. Numerical evolution confirms the late time asymptotic approximation and shows the expected effects of the cross parity pseudoscalar coupling.