The possibility that two toroidal accretion configurations may be orbiting around a super-massive Kerr black hole has been addressed. Such tori may be formed during different stages of the Kerr attractor accretion history. We consider the relative rotation of the tori and the corotation or counterrotation of a single torus with respect to the Kerr attractor. We give classification of the couples of accreting and non-accreting tori in dependence on the Kerr black hole dimensionless spin. We demonstrate that only in few cases a double accretion tori system may be formed under specific conditions.