  Most slowly synchronizing automata over binary alphabets are circular, i.e.,
containing a letter permuting the states in a single cycle, and their set of
synchronizing words has maximal state complexity, which also implies complete
reachability.Here, we take a closer look at generalized circular and completely
reachable automata. We derive that over a binary alphabet every completely
reachable automaton must be circular, a consequence of a structural result
stating that completely reachable automata over strictly less letters than
states always contain permutational letters. We state sufficient conditions for
the state complexity of the set of synchronizing words of a generalized
circular automaton to be maximal. We apply our main criteria to the family
$\mathscr K_n$ of automata that was previously only conjectured to have this
property.
