Considering the increasing coupling between transmission and distribution networks, an affine arithmetic-based interval power flow algorithm of integrated transmission and distribution (I-T&D) networks is proposed. First, a novel twostep approximation method for sine and cosine functions is proposed to reduce the number of noise variables in the affine form of the injected power, as the basis of the proposed algorithm. Then, this algorithm is divided into four stages, which are implemented alternatively between a transmission system operator (TSO) and multiple distribution system operators (DSOs) by exchanging parameters at TSO-DSO boundaries. Three-phase unbalanced I-T&D networks and boundary tap changer action are also discussed. In addition, an asynchronous power flow method is proposed to mitigate the influences of communication networks on the proposed interval power flow algorithm. Numerical experiments demonstrate the effectiveness of the proposed two-step approximation method, as well as the conservative property, fairly good approximation effect, high efficiency, and low data exchange amount of the proposed algorithm in various circumstances, including single-phase and three-phase networks, varying degrees of unbalance, various smart inverter controls, etc.