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In this paper, the inverse of a nonsingular, centroskewsymmetric Toeplitz-plus-Hankel Bezoutian B of (even) order n are computed, and a representation of B^(â1) as a sum of a Toeplitz and a Hankel matrix is found. Two possibilities are discussed. In the first one, the problem is reduced to the inversion of two skewsymmetric Toeplitz Bezoutians of order n. In the second one, the problem is tackled via the inversion of two Hankel Bezoutians of half the order n/2. The inversion of Toeplitz or Hankel Bezoutians is the subject of a previous paper [T. Ehrhardt and K. Rost. Resultant matrices and inversion of Bezoutians. Linear Algebra Appl., 439:621â639, 2013.]. Both approaches lead to fast O(n^2) inversion algorithms.