  We complete the analysis on the birational rigidity of quasismooth Fano
3-fold deformation families appearing in the Graded Ring Database as a complete
intersection. When such a deformation family $X$ has Fano index at least 2 and
is minimally embedded in a weighted projective space in codimension 2, we
determine which cyclic quotient singularity is a maximal centre. If a cyclic
quotient singularity is a maximal centre, we construct a Sarkisov link to a
non-isomorphic Mori fibre space or a birational involution. This allows, in
particular, the construction of new examples of Fano 3-folds of codimension 6
which are realised as complete intersections in fake weighted projective
spaces. We define linear cyclic quotient singularities on $X$ and prove that
these are maximal centres by explicitly computing Sarkisov links centred at
them. It turns out that each $X$ has a linear cyclic quotient singularity
leading to a new birational model. As a consequence, we show that if $X$ is
birationally rigid then its Fano index is 1. If the new birational model is a
strict Mori fibre space, we determine its fibration type explicitly. In this
case, a general member of $X$ is birational to a del Pezzo fibration of degrees
1, 2 or 3 or to a conic bundle $Y/S$ where $S$ is a weighted projective plane
with at most $A_2$ singularities.
