Let
{\mathcal{A}=\mathcal{A}_{p}}
be the
{\mathrm{mod}\,p}
Steenrod algebra, where p is a fixed prime and let
{\mathcal{A}^{\prime}}
denote the Bockstein-free part of
{\mathcal{A}}
at odd primes.
Being a connected graded Hopf algebra,
{\mathcal{A}}
has the canonical conjugation χ. Using this map, we introduce a relationship between the X- and Z-bases of
{\mathcal{A}^{\prime}}
.
We show that these bases restrict to give bases to the well-known sub-Hopf algebras
{\mathcal{A}(n-1)}
,
{n\geq 1}
, of
{\mathcal{A}^{\prime}}
.