Interferometers operating at or close to quantum limits of precision have found wide application in tabletop searches for physics beyond the standard model, the study of fundamental forces and symmetries of nature and foundational tests of quantum mechanics. The limits imposed by quantum fluctuations and measurement backaction on conventional interferometers (δϕ∼1/√N) have spurred the development of schemes to circumvent these limits through quantum interference, multiparticle interactions and entanglement. A prominent example of such schemes, the so-called SU(1,1) interferometer, has been shown to be particularly robust against particle loss and inefficient detection, and has been demonstrated with photons and ultracold atoms. Here, we realize a SU(1,1) interferometer in a fundamentally new platform in which the interfering arms are distinct flexural modes of a millimeter-scale mechanical resonator. We realize up to 15.4(3) dB of noise squeezing and demonstrate the Heisenberg scaling of interferometric sensitivity (δϕ∼1/N), corresponding to a 6-fold improvement in measurement precision over a conventional interferometer. Our work extends the optomechanical toolbox for the quantum manipulation of macroscopic mechanical motion and presents new avenues for studies of optomechanical sensing and the nonequilibrium dynamics of multimode optomechanical systems.