  We define and discuss a notion called fibered commensurability of outer
automorphisms of free groups. This notion lets us study symmetry of outer
automorphisms. The notion of fibered commensurability is first defined by
Calegari-Sun-Wang on mapping class groups. The Nielsen-Thurston type of mapping
classes is a commensurability invariant. One of the important facts of fibered
commensurability on mapping class groups is for the case of pseudo-Anosovs,
there is a unique minimal element in each fibered commensurability class. For
outer automorphisms, we first show that being atoroidal and fully irreducible
is a commensurability invariant. Then for such outer automorphisms, we prove
that there is a unique minimal element in each fibered commensurability class,
under a certain asymmetry condition on the ideal Whitehead graphs.
