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The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such linearizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In this context, Hermitian matrix polynomials are one of the most important classes of matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic is a set of signs attached to these real eigenvalues which is crucial for determining the behavior of systems described by Hermitian matrix polynomials and, therefore, it is desirable to develop linearizations that preserve the sign characteristic of these polynomials, but, at present, only one such linearization is known. In this paper, we present a complete characterization of all the Hermitian strong linearizations that preserve the sign characteristic of a given Hermitian matrix polynomial and identify several families of such linearizations that can be constructed very easily from the coefficients of the polynomial.