Excitons in monolayer semiconductors have large optical transition dipole for strong coupling with light field. Interlayer excitons in heterobilayers, with layer separation of electron and hole components, feature large electric dipole that enables strong coupling with electric field and exciton-exciton interaction, at the cost that the optical dipole is substantially quenched (by several orders of magnitude). In this letter, we demonstrate the ability to create a new class of excitons in transition metal dichalcogenide (TMD) hetero- and homo-bilayers that combines the advantages of monolayer- and interlayer-excitons, i.e. featuring both large optical dipole and large electric dipole. These excitons consist of an electron that is well confined in an individual layer, and a hole that is well extended in both layers, realized here through the carrier-species specific layer-hybridization controlled through the interplay of rotational, translational, band offset, and valley-spin degrees of freedom. We observe different species of such layer-hybridized valley excitons in different heterobilayer and homobilayer systems, which can be utilized for realizing strongly interacting excitonic/polaritonic gases, as well as optical quantum coherent controls of bidirectional interlayer carrier transfer either with upper conversion or down conversion in energy.