The authors obtain linear congruences for special values of the Kubota-Leopoldt 2-adic \(L\)-functions \(L_2(s,\chi)\). More exactly, they evaluate the 2-adic value of various linear combinations of the numbers \(L_2(k,\chi\omega^{1-k})\), where \(k\) runs through an arbitrary finite set of rational integers, \(\chi\) runs through a set of quadratic characters and \(\omega\) denotes the Teichmüller character mod 4. The result is, in the authors' words, the most general linear congruence relation of the type in question. It provides an immediate generalization for some previous results by the first author and also by \textit{A. Wójcik} [Compos. Math. 111, No. 3, 289--304 (1998; Zbl 0915.11052)]. A crucial step in the proof is based on divisibility properties of generalized Vandermonde determinants. This research has its origin in results giving linear congruences modulo a power of 2 between class numbers of imaginary quadratic fields; see, e.g., an article by \textit{K. Hardy} and \textit{K. S. Williams} [Acta Arith. 52, 263--276 (1986; Zbl 0557.12003)].