Wadler proposed the dual calculus, which corresponds to classical sequent calculus LK, and studied the relationship between the λμ-calculus and the dual calculus as equational systems to explain the duality between call-by-value and call-by-name in a purely syntactical way. Wadler left an open question whether one can obtain similar results by replacing the equations with reductions. This paper gives one answer to his question. We first refine the λμ-calculus as reduction systems by reformulating sum types and omitting problematic reduction rules that are not simulated by reductions of the dual calculus. Secondly, we give translations between the call-by-name λμ-calculus and the call-by-name dual calculus, and show that they preserve the call-by-name reductions. We also show that the compositions of these translations become identity maps up to the call-by-name reductions. We also give translations for the call-by-value systems, and show that they satisfy properties similar to the call-by-name translations. Thirdly, we introduce translations between the call-by-value λμ-calculus and the call-by-name one by composing the above translations with duality on the dual calculus. We finally obtain results corresponding to Wadler's, but our results are based on reductions.