Summary: A quasigroup is said to be universal relative to a property if such a property is isotopic invariant in quasigroups. An \(m\)-inverse quasigroup is called an odd \(m\)-inverse quasigroup if \(m\) is an odd integer. This study presents a special type of middle isotopism (called the \(\mathcal T_1\) condition) under which odd \(m\)-inverse quasigroups are universal. A sufficient condition for an odd \(m\)-inverse quasigroup that is specially isotopic to a quasigroup to be isomorphic to the quasigroup isotope is established. It is shown that under this special type of middle isotopism, if \(n\) is a positive even integer and \(m\) is a positive odd integer, then, a quasigroup is an odd \(m\)-inverse quasigroup with an inverse cycle of length \(nm\) if and only if its quasigroup isotope is an odd \(m\)-inverse quasigroup with an inverse cycle of length \(nm\). But when \(n\) is a positive odd integer, if a quasigroup is an odd \(m\)-inverse quasigroup with an inverse cycle of length \(nm\), its quasigroup isotope is an odd \(m\)-inverse quasigroup with an inverse cycle of length \(nm\) if and only if the two quasigroups are isomorphic. Hence, they are isomorphic odd \(m\)-inverse quasigroups. Some examples of middle isotopic odd \(m\)-inverse quasigroups with the \(\mathcal T_1\) condition are used to illustrate the results above. Explanations and procedures are given on how these results can be used to apply odd \(m\)-inverse quasigroups to cryptography.