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In this paper we consider Q-Fano 3-fold weighted complete intersections ofcodimension 2 in the 85 families listed in the Iano-Fletcher's list anddetermine which cycle is a maximal center or not. For each maximal center, weconstruct either a birational involution which untwists the maximal singularityor a Sarkisov link centered at the cycle to an another explicitly describedMori fiber space. As a consequence, 19 families are proved to be birationallyrigid and the remaining 66 families are proved to be birationally nonrigid.

Birational Mori fiber structures of Q-Fano 3-fold weighted complete intersections