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