We analyze the spaces of trivariate C 1-smooth isogeometric functions on two-patch domains. Our aim is to generalize the corresponding results from the bivariate [25] to the trivariate case. In the first part of the paper, we introduce the notion of gluing data and use it to define glued spline functions on two-patch domains. Applying the fundamental observation that "matched G k-constructions always yield C k-continuous isogeometric elements", see [14], to graph hypersurfaces in four-dimensional space, allows us to characterize C 1-smooth geometrically continuous isogeometric functions as the push-forwards of these functions for suitable gluing data. The second part of the paper is devoted to various special classes of gluing data. We analyze how the generic dimensions depend on the number of knot spans (elements) and on the spline degree. Finally we show how to construct locally supported basis functions in specific situations.