Reflectometry is the task for acquiring the bidirectional reflectance distribution function (BRDFs) of real-world materials. The typical reflectometry pipeline in computer vision, computer graphics, and computational imaging involves capturing images of a convex shape under multiple illumination and imaging conditions; due to the convexity of the shape, which implies that all paths from the light source to the camera perform a single reflection, the intensities in these images can subsequently be analytically mapped to BRDF values. We deviate from this pipeline by investigating the utility of higher-order light transport effects, such as the interreflections arising when illuminating and imaging a concave object, for reflectometry. We show that interreflections provide a rich set of contraints on the unknown BRDF, significantly exceeding those available in equivalent measurements of convex shapes. We develop a differentiable rendering pipeline to solve an inverse rendering problem that uses these constraints to produce high-fidelity BRDF estimates from even a single input image. Finally, we take first steps towards designing new concave shapes that maximize the amount of information about the unknown BRDF available in image measurements. We perform extensive simulations to validate the utility of this reflectometry from interreflections approach.