Ergodic properties of skew-extensions of irrational rotations (Anzai products) are studied. Several new invariants are introduced and examined. There are useful to distinguish between absolutely continuous cocycles and those given by step functions. The roles of diophantine approximation properties for the underlying rotation and of the topological degree of absolutely continuous cocycles are exhibited. In particular, no absolutely continuous cocycles with nonzero degree is a coboundary. The set of ergodic self-joinings of an automorphism of a Lebesgue space is considered and related invariants are introduced. They are used in particular to classify skew-product extensions of rotations. Applications to Anzai products and related questions on the existence of coalescent infinite self-joinings are examined.