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We study n-length consta-Abelian codes (a generalization of the well-known Abelian codes and constacyclic codes) over Galois rings of characteristic pa, where n and p are coprime. A twisted discrete Fourier transform (DFT) is used to generalize transform domain results of Abelian and constacyclic codes, to consta-Abelian codes. Further, we characterize consta-Abelian codes invariant under two kinds of monomials, whose underlying permutations are effected by: i) multiplying the coordinates with a unit in the appropriate mixed-radix representation of the coordinate positions and ii) shifting the coordinates by t positions. All the codes studied here belong to the class of quasi-twisted codes which are known to contain some good codes. We show that the dual of a consta-Abelian code invariant under the two monomials is also a consta-Abelian code closed under both monomials.