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The increasing interest in spin-based electronics has led to a vigorous search for new materials that can provide a high degree of spin polarization in electron transport. An ideal candidate would act as an insulator for one spin channel and a conductor or semiconductor for the opposite spin channel, corresponding to the respective cases of half-metallicity and spin-gapless semiconductivity. Our first-principle electronic-structure calculations indicate that the metallic Heusler compound Ti₂MnAl becomes half-metallic and spin-gapless semiconducting if half of the Al atoms are replaced by Sn and In, respectively. These electronic structures are associated with structural transitions from the regular cubicmore » Heusler structure to the inverted cubic Heusler structure.« less