A reduced-gravity (1$frac{1}{2}$-layer) model forced by daily climatological winds simulates twin, anticyclonic gyres, which propagate westward on either side of the equator. The gyres form at the beginning of both the Southwest Monsoon and the Northeast monsoon in the equatorial eastern Indian Ocean, and subsequently propagate across the basin. Their existence is supported by velocity observations taken during WOCE in 1995 and by TOPEX/Poseidon sea-level observations during 1993. They are also present in the ECCO model/data product. They form at the front of a Rossby-wave packet generated by the reflection of the equatorial jet (EJ) from the eastern boundary of the basin. They are likely either Rossby solitons or result from the nonlinear interaction between the EJ and the Rossby-wave front.