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Linear channel equalization in block transmission systems amounts to inverting Toeplitz systems of linear equations. Motivated by limitations of a previous blind block equalizer, we derive properties and investigate the class of tall Toeplitz matrix inverses which themselves exhibit (even approximate) Toeplitz structure. The class is characterized by the size of leading and trailing all-zero block submatrices, and interesting links as well as optimal choices of the size parameter are established with the number of minimum-phase zeros of the underlying channel transfer function. Exploiting the properties of such equalizers we derive a direct blind adaptive equalizer and illustrate superiority over competing approaches. It is also shown that the optimum delay for blind block equalization corresponds to the number of maximum-phase channel zeros.