Axion-like particles are a broad class of dark matter candidates which are expected to behave as a coherent, classical field with a weak coupling to photons. Research into the detectability of these particles with laser interferometers has recently revealed a number of promising experimental designs. Inspired by these ideas, we propose the Axion Detection with Birefringent Cavities (ADBC) experiment, a new axion interferometry concept using a cavity that exhibits birefringence between its two, linearly polarized laser eigenmodes. This experimental concept overcomes several limitations of the designs currently in the literature, and can be practically realized in the form of a simple bowtie cavity with tunable mirror angles. Our design thereby increases the sensitivity to the axion-photon coupling over a wide range of axion masses.