Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation A : R[t] -> R[t] defined by A(t(n)) = A(n)(t), where A(n)(t) denotes the n-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator A, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials.