  We prove that the property of a free group endomorphism being irreducible is
a group invariant of the ascending HNN extension it defines. This answers a
question posed by Dowdall-Kapovich-Leininger. We further prove that being
irreducible and atoroidal is a commensurability invariant. The invariance
follows from an algebraic characterization of ascending HNN extensions that
determines exactly when their defining endomorphisms are irreducible and
atoroidal; specifically, we show that the endomorphism is irreducible and
atoroidal if and only if the ascending HNN extension has no infinite index
subgroups that are ascending HNN extensions.
