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Two elements g, h of a permutation group G acting on a set V are said to be intersecting if g (v) = h (v) for some v ∈ V . More generally, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection density ρ (G) of a transitive permutation group G is the maximum value of the quotient | F | / | G v | where F runs over all intersecting sets in G and G v is the stabilizer of v ∈ V . A vertex-transitive graph X is intersection density stable if any two transitive subgroups of Aut (X) have the same intersection density. This paper studies the above concepts in the context of cubic symmetric graphs. While a 1-regular cubic symmetric graph is necessarily intersection density stable, the situation for 2-arc-regular cubic symmetric graphs is more complex. A necessary condition for a 2-arc-regular cubic symmetric graph admitting a 1-arc-regular subgroup of automorphisms to be intersection density stable is given, and an infinite family of such graphs is constructed. [ABSTRACT FROM AUTHOR]