  We investigate the natural codings of linear involutions. We deduce from the
geometric representation of linear involutions as Poincar\'e maps of measured
foliations a suitable definition of return words which yields that the set of
first return words to a given word is a symmetric basis of the free group on
the underlying alphabet $A$. The set of first return words with respect to a
subgroup of finite index $G$ of the free group on $A$ is also proved to be a
symmetric basis of $G$.
