  We define the concept of \emph{companion automorphism} of a Hopf algebra $H$
as an automorphism $\sigma:H \rightarrow H$: $\sigma^2=S^2$ --where $S$ denotes
the antipode--. A Hopf algebra is said to be \emph{almost involutive} (AI) if
it admits a companion automorphism that can be viewed as a special additional
symmetry. We present examples and study some of the basic properties and
constructions of AI-Hopf algebras centering the attention in the finite
dimensional case. In particular we show that within the family of Hopf algebras
of dimension smaller or equal than 15, only in dimension eight and twelve,
there are non almost involutive Hopf algebras.
