The Wiener-Hopf factorization of a \(2\times 2\) matrix function \(G\) defined on a closed Carleson curve \(\Gamma\) is considered. Two types of transformation of \(G\) are discussed, namely, \[ G \mapsto^{G} = U G V,\tag{1} \] where \(U, V\) are rational \(2\times 2\) matrix functions with no poles on \(\Gamma\), and \[ G \mapsto^{G} = U_{-} G V_{+},\tag{2} \] where \(U_{-}, V_{+}\) are rational \(2\times 2\) matrix functions with no poles on \(\Gamma \bigcup \operatorname{ext} \Gamma\), and \(\Gamma \bigcup \operatorname{int} \Gamma\) respectively. A classification scheme based on these transformations is established. Invariants for (1) and (2) are found. Conditions on a matrix function \(G\in L^{\infty}(\Gamma)^{2\times 2}\) are determined under which the matrix function \(G\) can be transformed by (1) and (2) (or by some other rational transformations) into triangular or Daniel-Khrapkov form.