A groupoid term is said to be linear if every variable has at most one occurrence in it. A \(\star\)-linear equational theory is an equational theory \(E\) such that every term \(t\) is equivalent to a unique linear term \(t^{\star}\). The authors prove that:
– there are precisely six \(\star\)-linear equational theories of groupoids;
– four of them have a finite equational base, and the other two are inherently non-finitely based.
Also, all subvarieties of the six corresponding varieties are described and small generating groupoids for each of them are presented.