The class of all regular semigroups with inverse transversals is closed under direct products, IST-subsemigroups and homomorphic images. It forms an IST-variety IST\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{IST}$$\end{document}. The IST-variety is different from not only the regular unary semigroup varieties, but also the regular semigroup e-varieties. IST\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{IST}$$\end{document} contains many IST-varieties as its subvarieties. For example, it contains the IST-variety of all orthodox semigroups with inverse transversals OIST\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{OIST}$$\end{document}, the IST-variety of all (left regular, right regular) bands with inverse transversals (LRBIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{LRBIT}$$\end{document}, RRBIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{RRBIT}$$\end{document}) BIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{BIT}$$\end{document}, the IST-variety of all regular semigroups with Q-inverse transversals QIST\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{QIST}$$\end{document}, etc. According to the descriptions for the free objects in LRBIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{LRBIT}$$\end{document}, RRBIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{RRBIT}$$\end{document} and BIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{BIT}$$\end{document}, we characterize further in this paper the words in the following free IST-semigroups: the free IST-bands in BIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{BIT}$$\end{document}, the free left regular IST-bands in LRBIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{LRBIT}$$\end{document} and the free right regular IST-bands in RRBIT\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{RRBIT}$$\end{document} respectively. Classifications of Blyth for regular semigroups with inverse transversals and characterizations of Gerhardt for words on free bands are hence generalized and enriched.
