This paper studies the classification of recursive sets by the number of tape reversals required for their recognition on a two-tape Turing machine with a one-way input tape. This measure yields a rich hierarchy of tape reversal limited complexity classes and their properties and ordering are investigated. The most striking difference between this and the previously studied complexity measures lies in the fact that the "speed-up" theorem does not hold for slowly growing tape reversal complexity classes. These differences are discussed, and several relations between the different complexity measures and languages are established.