The paper deals with oscillatory criteria for the solutions of a scalar higher order neutral equation of mixed type (with delayed and advanced arguments), with constant coefficients. As a main result, the classical characterization of oscillations by the characteristic equation is extended to such equations. Extension is from the delay case (which can be deduced from earlier papers) to the mixed type case. Two methods have been developed during the last decade in the study of oscillations: 1) Historically the first one is what can be called a direct method. Amongst those who have contributed to its development, one can quote the three authors, and also, with no attempt to be complete, \textit{G. Ladas} and \textit{Y. G. Sficas} (Numerous references to related works are given in the paper). This is the method chosen here. 2) The second method is based on the Laplace transform. It was introduced in a joint paper by \textit{I. Györi} and the reviewer [J. Differ. Equations 81, 98-105 (1989; Zbl 0691.34054)]. Incidentally, the same result as the one under review was proved by means of the Laplace transform method by \textit{M. A. El Attar} and the reviewer in a joint work which appeared this year (in the Proceedings of the Royal Society of Edinburgh, 124 A, pp. 909-916).