Coordinated cell rotation along a curved matrix interface can sculpt epithelial tissues into spherical morphologies. Subsequently, radially-oriented invasion of multicellular strands or branches can occur by local remodeling of the confining matrix. These symmetry-breaking transitions emerge from the dynamic reciprocity between cells and matrix, but remain poorly understood. Here, we show that epithelial cell spheroids collectively transition from circumferential orbiting to radial invasion via bi-directional interactions with the surrounding matrix curvature. Initially, spheroids exhibit an ellipsoidal shape but become rounded as orbiting occurs. However, cells gradually reorient from coordinated rotation towards outward strand invasion due to the accumulation of contractile tractions at discrete sites. Remarkably, the initial ellipsoid morphology predicts subsequent invasion of 2-4 strands roughly aligned with the major axis. We then perturb collective migration using osmotic pressure, showing that orbiting can be arrested and invasion can be reversed. We also investigate coordinated orbiting in "mosaic" spheroids, showing a small fraction of "leader" cells with weakened cell-cell adhesions can impede collective orbiting but still invade into the matrix. Finally, we establish a minimal self-propelled particle model to elucidate how collective orbiting is mediated by the crosstalk of cell-cell and cell-matrix adhesion along a curved boundary. Altogether, this work elucidates how tissue morphogenesis is governed by the interplay of collective behaviors and the local curvature of the cell-matrix, with relevance for embryonic development and tumor progression.