Effects of magnetic asymmetry on strongly coupled spin-vortex pairs with parallel core polarization and antiparallel chirality in synthetic nanomagnets are investigated. This includes vortex-core length asymmetry, biasing field asymmetry, and pinning of one of the two vortex cores. Our experimental observations as well as analytical and micromagnetic modeling show how magnetic asymmetry can be used to differentiate magneto-resistively otherwise degenerate multiple stable states of a vortex pair. These results expand the knowledge base for spin vortex arrays in nanostructures and should be useful in light of the recent proposals on coding information into multiple topological spin states, such as single and multiple vortex core/chirality states.