The cumulative entropy is an information measure which is alternative to the differential entropy and is connected with a notion in reliability theory. Indeed, the cumulative entropy of a random lifetime $X$ can be expressed as the expectation of its mean inactivity time evaluated at $X$. After a brief review of its main properties, in this paper we relate the cumulative entropy to the cumulative inaccuracy, and provide some inequalities based on suitable stochastic orderings. We also show a characterization property of the dynamic version of the cumulative entropy. In conclusion, a stochastic comparison between the empirical cumulative entropy and the empirical cumulative inaccuracy is investigated.