Integrated quantum optics has the potential to markedly reduce the footprint and resource requirements of quantum information processing systems, but its practical implementation demands broader utilization of the available degrees of freedom within the optical field. To date, integrated photonic quantum systems have primarily relied on path encoding. However, in the classical regime, the transverse spatial modes of a multi-mode waveguide have been easily manipulated using the waveguide geometry to densely encode information. Here, we demonstrate quantum interference between the transverse spatial modes within a single multi-mode waveguide using quantum circuit-building blocks. This work shows that spatial modes can be controlled to an unprecedented level and have the potential to enable practical and robust quantum information processing.