Solid solutions InMn1-xGaxO3 (0 <= x <= 1) have been investigated using magnetic, dielectric, specific heat, differential scanning calorimetry (DSC), and high-temperature powder synchrotron X-ray diffraction (HT-SXRD) measurements. It was found that samples with 0.5 <= x <= 1 crystallize in space group P6(3)/mmc with a similar to 3.32 angstrom and c similar to 11.9 angstrom, and samples with 0.0 <= x <= 0.4 crystallize in space group P6(3)cm with a similar to 5.8 angstrom and c similar to 11.6 angstrom at room temperature. HT-SXRD data revealed the existence of a P6(3)cm-to-P6(3)/mmc phase transition at about 480 K in InMn0.6Ga0.4O3 and at 950 K in InMn0.7Ga0.3O3. However, no dielectric, phonon, second-harmonic-generation, or DSC anomalies were found to be associated with these phase transitions. The phase transition should be improper ferroelectric from the symmetry point of view, but the above-mentioned experimental facts, together with the absence of ferroelectric hysteresis loops, revealed no evidence for ferroelectricity in the low-temperature P63cm structure. We suggest that InMn1-xGaxO3 corresponds to a nonferroelectric phase of hexagonal RMnO3 with P6(3)cm symmetry. The antiferromagnetic phase-transition temperature decreases from 118 K for x = 0 to 105 K for x = 0.1 and 73 K for x = 0.2, and no long-range magnetic ordering could be found for x >= 0.3. Specific heat anomalies associated with short-range magnetic ordering were observed for 0.0 <= x <= 0.5. InMn1-xGaxO3 with small Mn contents (0.8 <= x <= 0.98) has a bright-blue color.