In combinatorics on words, the well-studied factor complexity function \(\rho _{\textbf{x}}\) of a sequence \(\textbf{x}\) over a finite alphabet counts, for every nonnegative integer n, the number of distinct length-n factors of \(\textbf{x}\) . In this paper, we introduce the reflection complexity function \(r_{\textbf{x}}\) to enumerate the factors occurring in a sequence \(\textbf{x}\) , up to reversing the order of symbols in a word. We prove a number of results about the growth properties of \(r_{\textbf{x}}\) and its relationship with other complexity functions. We also prove a Morse–Hedlund-type result characterizing eventually periodic sequences in terms of their reflection complexity, and we deduce a characterization of Sturmian sequences. We investigate the reflection complexity of quasi-Sturmian, episturmian, \((s+1)\) -dimensional billiard, complementation-symmetric Rote, and rich sequences. Furthermore, we prove that if \(\textbf{x}\) is k-automatic, then \(r_{\textbf{x}}\) is computably k-regular, and we use the software Walnut to evaluate the reflection complexity of some automatic sequences, such as the Thue–Morse sequence. We note that there are still many unanswered questions about this reflection measure.
