We report a 3D MHD simulation study of the interactions between radio galaxies (RGs) and galaxy-cluster-media shocks in which the incident shock normals are orthogonal to the bipolar active galactic nucleus (AGN) jets. Before shock impact, light, supersonic jets inflate lobes (cavities) in a static, uniform intracluster medium. We examine three AGN activity scenarios: (1) continued, steady jet activity; (2) jet source cycled off coincident with shock/radio lobe impact; (3) jet activity ceased well before shock arrival (a "radio phoenix" scenario). The simulations follow relativistic electrons (CRe) introduced by the jets, enabling synthetic radio synchrotron images and spectra. Such encounters can be decomposed into an abrupt shock transition and a subsequent long-term postshock wind. Shock impact disrupts the preformed, low-density RG cavities into two ring vortices embedded in the postshock wind. Dynamical processes cause the vortex pair to merge as they propagate downwind somewhat faster than the wind itself. When the AGN jets remain active, ram pressure bends the jets downwind, generating a narrow angle tail morphology aligned with the axis of the vortex ring. The deflected jets do not significantly alter dynamical evolution of the vortex ring. However, active jets and their associated tails do dominate the synchrotron emission, compromising the observability of the vortex structures. Downwind-directed momentum concentrated by the jets impacts and alters the postencounter shock. In the "radio phoenix" scenario, no DSA of the fossil electron population is required to account for the observed brightening and flattening of the spectra; adiabatic compression effects are sufficient.