Within a theoretical framework that accounts for all open-charm mesons, including the D-0*(2308), the D-sJ*(2317), and the D-sJ(2460), we analyze the structure and explore possible quantum number assignments for the D-sJ(2860) and the D-sJ(2700) mesons reported by BABAR and Belle Collaborations. The open-charm sector is properly described if considered as a combination of conventional quark-antiquark states and four-quark components. All negative parity and 2(+) states can be understood in terms only of q (q) over bar components, however the description of the 0(+) and 1(+) mesons is improved whenever the mixing between two- and four-quark configurations is included. We analyze all possible quantum number assignments for the D-sJ(2860) in terms of both c (s) over bar and cn (s) over tilde (n) over bar configurations. We discuss the role played by the electromagnetic and strong decay widths as basic tools to distinguish among them. The broad structure reported by BABAR near 2.7 GeV is also analyzed.