Excitons in monolayer semiconductors have a large optical transition dipole for strong coupling with light. Interlayer excitons in heterobilayers feature a large electric dipole that enables strong coupling with an electric field and exciton-exciton interaction at the cost of a small optical dipole. We demonstrate the ability to create a new class of excitons in hetero- and homobilayers that combines advantages of monolayer and interlayer excitons, i.e., featuring both large optical and electric dipoles. These excitons consist of an electron confined in an individual layer, and a hole extended in both layers, where the carrier-species-dependent layer hybridization can be controlled through rotational, translational, band offset, and valley-spin degrees of freedom. We observe different species of layer-hybridized valley excitons, which can be used for realizing strongly interacting polaritonic gases and optical quantum controls of bidirectional interlayer carrier transfer.