In previous work the author constructed a convolution algebra and an isomorphic multiplication algebra of one-dimensional associated homogeneous distributions with support in R. In this paper we investigate the various algebraic substructures that can be identified in these algebras. Besides identifying ideals and giving polynomial representations for six subalgebras, it is also shown that both algebras contain an interesting Abelian subgroup, which can be used to construct generalized integration/derivation operators of complex degree on the whole line R.