Using a recursive MDS matrix over a finite field in the diffusion layer of block ciphers is a standard method to maximize the diffusion while minimizing the implementation cost. On the other hand, an involutory MDS matrix (the matrix is the inverse of itself) simplifies the implementation of the cipher as the same module can be used for both encryption and decryption. However, it has been shown (Gupta et al.’21) that there doesn’t exist a recursive MDS matrix M that is also involutory over a field of charateristic 2 except for a trivial case. Another option is to use a recursive-like involutory MDS matrix of the form RM, where M is a recursive matrix and R is a permutation matrix. In fact, Gupta et al. proved that over a field of characteristic 2, if f is a monic, self-reciprocal polynomial of degree m, Cf is the companion matrix of f and M = Cfm is MDS, then RM is an involuntary MDS matrix, where R is the matrix reversing the order of rows in M. With RM in the diffusion layer, one can use the same LFSR for both encryption and decryption while still taking advantage of the fact that M is recursive. In this paper, we extend the work of Gupta et al. and establish a necessary and sufficient condition for the generator polynomials of Reed-Solomon codes over a finite field of arbitrary characteristic to be self-reciprocal, from which the desired RM matrices can be efficiently constructed for all feasible parameters.