We describe a new type of interferometer, the coupled spiral interferometer (CSPIN), which consists in a planar spiral-shaped waveguide with optical coupling between adjacent arms. We demonstrate through numerical simulations that when the arms are phase matched and contain straight parallel sections, a CSPIN supports strong slow-light resonances that are periodic in frequency and in the coupling coefficient of the arms. Compared to existing coupled resonator optical waveguides (CROWs), it presents the significant benefit of greater compactness (as 1/N, where N is the number of coupled resonators, or equivalently the number of arms). Being two dimensional, it is also easier to fabricate than a coil optical resonator (COR). We illustrate this concept with a spiral in the shape of a square with rounded corners and two arms. Resonant conditions in wavelength and coupling coefficient are shown to exist for which the group delay goes to infinity and the transmission goes to zero, as in a COR. The effective group delay (product of group delay and transmission), a useful figure of merit to quantify the efficacy with which light can be slowed down, is maximum on both sides of a wavelength resonance. For a square spiral ~70 mm in length made of a low-loss (0.16 m-1) silicon nitride waveguide, the maximum effective group delay is 15.9 ns (or a slow-down factor of 34.6). This value is independent of the number of arms (for a small loss), and it is the same value as for a COR with the same length and loss (for any N) and for a CROW with two resonators. This study shows that CSPINs offer a wide range of exciting new opportunities for implementing and utilizing slow light in numerous applications on a highly compact platform.