Existing theoretical models of interdependent networks are mostly aimed at complex network systems which composed of undirected subnets. However, in practical scenarios, most interdependent network systems are formed by coupling directed subnets and undirected subnets (e.g., physical supply chain networks coupled with information networks, transportation networks coupled with communication networks). Therefore, in this work, we focus on this ubiquitous type of interdependent network: the directed-undirected interdependent network, and research the cascading failure of directed-undirected interdependent networks with different coupling modes. We first establish a degree-based cascading model to characterize the cascading failure process of the directed-undirected interdependent networks. Considering the diversity of coupling modes in practical interdependent networks, we generate different types of directed-undirected interdependent networks with varying coupling modes (i.e., one-to-one, one-to-many, and many-to-one) and investigate the cascading failure robustness of these types of networks. Finally, through extensive simulation experiments, the cascading failure robustness of the networks under single-node attacks and multi-node attacks are explored. Some conclusions are summarized after this research: 1) The directed-undirected interdependent network robustness is positively related to the tolerance coefficient and exponential coefficient; 2) High degree nodes and high in-degree nodes should be protected to improve the robustness of the directed-undirected interdependent network; 3) The more interdependent edges the nodes of the attacked network have that point to another layer of the network, the less robust the interdependent network is.