Topological deep learning (TDL) is a rapidly growing field that seeks to leverage topological structure in data and facilitate learning from data supported on topological objects, ranging from molecules to 3D shapes. Most TDL architectures can be unified under the framework of higher-order message-passing (HOMP), which generalizes graph message-passing to higher-order domains. In the first part of the paper, we explore HOMP's expressive power from a topological perspective, demonstrating the framework's inability to capture fundamental topological and metric invariants such as diameter, orientability, planarity, and homology. In addition, we demonstrate HOMP's limitations in fully leveraging lifting and pooling methods on graphs. To the best of our knowledge, this is the first work to study the expressivity of TDL from a topological perspective. In the second part of the paper, we develop two new classes of architectures -- multi-cellular networks (MCN) and scalable MCN (SMCN) -- which draw inspiration from expressive GNNs. MCN can reach full expressivity, but scaling it to large data objects can be computationally expansive. Designed as a more scalable alternative, SMCN still mitigates many of HOMP's expressivity limitations. Finally, we design new benchmarks for evaluating models based on their ability to learn topological properties of complexes. We then evaluate SMCN on these benchmarks as well as on real-world graph datasets, demonstrating improvements over both HOMP baselines and expressive graph methods, highlighting the value of expressively leveraging topological information.