Microwave vision from spaceborne synthetic aperture radar (SAR) provides an all-weather, day-and-night capability to observe Earth, yet much of the information encoded in its signals remains undeciphered. Recent high-resolution imagery has revealed a striking phenomenon: man-made structures systematically appear in a spectrum of colours, the physical origin of which has been an open question. Here we show that this effect, which we term the microwave rainbow, is a form of geometric dispersion arising from structures acting as intrinsic diffraction gratings. We introduce a geometric-physical model that provides a direct analytical link between a target's geometry and its observed colour signature. This model quantitatively explains the full range of signatures, from continuous colour gradients on curved surfaces (zero-order diffraction) to repeating spectral patterns from periodic structures (high-order diffraction). This work transforms colour from a visual artefact into a precise measure of physical form, enabling the geometry of both critical infrastructure and natural phenomena to be mapped directly from space. Our findings establish the physical basis for a new remote sensing modality: microwave colour vision, and open a new frontier in how we perceive our world.