Proper two-sided exits of a Lévy process

It is proved that the two-sided exits of a Lévy process are proper, i.e. are not a.s. equal to their one-sided counterparts, if and only if said process is not a subordinator or the negative of a subordinator. Furthermore, Lévy processes are characterized, for which the supports of the first exit times from bounded annuli, simultaneously on each of the two events corresponding to exit at the lower and the upper boundary, respectively are unbounded, contain 0, are equal to [0,∞).