Summary: This is a study on the class of \(\text{FIM}(X)\)-languages and its important subfamily consisting of inverse automata languages (\(i\)-languages). Both algebraic and combinatorial approaches are used to obtain several results concerning closure operators on \((X\cup X^{-1})^*\)-languages, including a classification of \(\text{FIM}(X)\)-languages by \(i\)-languages. In particular, it is proved that the \(i\)-closure of a recognizable \((X\cup X^{-1})^*\)-language is at most deterministic context-free. Infinite trees are an essential tool in this process, and they are also helpful in producing counterexamples for other closure problems. Applications to \(X^*\)-languages are also produced, involving particular classes of codes.