Summary: The variety of cyclic Boolean algebras is a particular subvariety of the variety of tense algebras. The objective of this paper is to study the variety \({\mathcal T}\) of \(\{\to,g,h\}\)-subreducts of cyclic Boolean algebras, which we call cyclic Tarski algebras. We prove that \({\mathcal T}\) is generated by its finite members, and we characterise the locally finite subvarieties of \({\mathcal T}\). We prove that there are no splitting varieties in the lattice \(\Lambda({\mathcal T})\) of subvarieties of \({\mathcal T}\). Finally, we prove that the subquasivarieties and the subvarieties of a locally finite subvariety of \({\mathcal T}\) coincide.