Wave-domain processing is an emerging paradigm where signal processing operations are partially shifted from the digital to the electromagnetic (EM) domain. Leveraging reconfigurable EM devices, this approach aims to reduce complexity, energy consumption, and latency in next-generation wireless systems employing holographic MIMO. This paper establishes fundamental theorems on the controllability of generic reconfigurable EM devices, where wave processing is achieved through the dynamic configuration of passive scatterers. Specifically, we derive necessary and sufficient conditions for controllability as a function of geometry and mutual coupling between elements. Finally, we provide a detailed discussion and numerical results characterizing the interplay between the number of elements, physical size, degrees of freedom, and directivity.