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We explore the accretion properties of the black hole X-ray binary \j1550\ during its outbursts in 1998/99 and 2000. We model the disk, corona, and reflection components of X-ray spectra taken with the {\it Rossi X-ray Timing Explorer} (\rxte), using the {\tt relxill} suite of reflection models. The key result of our modeling is that the reflection spectrum in the very soft state is best explained by disk self-irradiation, i.e., photons from the inner disk are bent by the strong gravity of the black hole, and reflected off the disk surface. This is the first known detection of thermal disk radiation reflecting off the inner disk. There is also an apparent absorption line at $\sim6.9$ keV which may be evidence of an ionized disk wind. The coronal electron temperature ($kT_{\rm e}$) is, as expected, lower in the brighter outburst of 1998/99, explained qualitatively by more efficient coronal cooling due to irradiating disk photons. The disk inner radius is consistent with being within a few times the innermost stable circular orbit (ISCO) throughout the bright-hard-to-soft states (10s of $r_{\rm g}$ in gravitational units). The disk inclination is low during the hard state, disagreeing with the binary inclination value, and very close to $90^{\circ}$ in the soft state, recovering to a lower value when adopting a blackbody spectrum as the irradiating continuum.