<p>Linear intersection pairs of linear codes have become of interest due to their nice algebraic properties and wide applications. In this paper, we focus on linear intersection pairs of cyclic codes over finite fields. Some properties of cyclotomic cosets in cyclic groups are presented as key tools in the study of such linear intersection pairs. Characterizations and constructions of two cyclic codes of a fixed intersecting dimension are given in terms of their generator polynomials and cyclotomic cosets. In some cases, constructions of two cyclic codes of a fixed intersecting subcode are presented as well. Based on the theoretical characterization, some numerical examples of linear intersection pairs of cyclic codes with good parameters are illustrated.</p>