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The standard way to solve polynomial eigenvalue problems P(lambda)x=0 is to convert the matrix polynomial P(lambda) into a matrix pencil that preserves its spectral information — a process known as linearization. When P(lambda) is palindromic, the eigenvalues, elementary divisors, and minimal indices of P(lambda) have certain symmetries that can be lost when using the classical first and second Frobenius companion linearizations for numerical computations, since these linearizations do not preserve the palindromic structure. Recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original P(lambda) might possess, with particular attention to the preservation of palindromic structure. However, no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of linearizations for odd degree polynomials P(lambda) which are palindromic whenever P(lambda) is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family.