We investigate the effect of phase inversion on the domain morphology and theological properties of phase-separated polybutadiene/polyisoprene blends using optical microscopy, light scattering, and rheometry. Two different blends, low-vinyl polybutadiene/low-vinyl polyisoprene (LPB/LPI) and low-vinyl polybutadiene/high-vinyl polyisoprene (LPB/HPI), were used in this study. The LPB/LPI blend has a lower critical solution temperature (LCST) of (62 +/- 1) degreesC, while the LPB/HPI blend exhibits upper-critical-solution-temperature (UCST) behavior with a critical temperature above the experimentally accessible temperature window. We determine the quiescent phase-inversion composition (phi (LPI) = 0.55 +/-0.05) of phase-separated LPB/LPI blends from the discontinuity in the dynamic storage modulus and shear viscosity. For LPB/HPI, we find that a shear-induced coexisting structure (apparent as a "walnutlike" light-scattering pattern) develops at a fixed composition (phi (HPI) = 0.8) and constant temperature. The coexisting morphology consists of two different anisotropic structures; stringlike domains and small(xi < 5 mum) vorticity-aligned domains. We suggest that Shear-induced phase inversion may lie at the foundation of this effect, although more theoretical and experimental work is needed to verify this.