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The concepts of a dihedral and a reflexive module with $\infty$-simplicialfaces are introduced. For each involutive $A_\infty$-algebra, the dihedral andthe reflexive tensor modules with $\infty$-simplicial faces are constructed. Onthe basis of dihedral and reflexive modules with $\infty$-simplicial faces thatdefined by an involutive $A_\infty$-algebra the constructions of the dihedraland the reflexive homology of involutive $A_\infty$-algebras over any unitalcommutative rings are given. The conception of an involutive homotopy unital$A_\infty$-algebra is introduced. A long exact sequence that connecting thedihedral and the reflexive homology of involutive homotopically unital$A_\infty$-algebras over any unital commutative rings is constructed.

Dihedral and reflexive modules with $\infty$-simplicial faces and  dihedral and reflexive homology of involutive $A_\infty$-algebras over unital  commutative rings