The effects of travel in the Janus Vortex mirror the original transporter accident that transported James T. Kirk, Leonard McCoy, Montgomery Scott and Nyota Uhura to the mirror universe in 2267. The two universes ran parallel in terms of the fact that both versions of the Excelsior were travelling across the vortex from opposite ends. From the perspective of the crew, their ship changed to the ISS Excelsior, their uniforms had changed and a scar appeared on Captain Hikaru Sulu's face. This suggests that at some point within the anomaly, both ships exist as one, and when they separate, only brain patterns exchange places rather than physical matter. (TOS video game: Shattered Universe)
Upon arrival in the universe, Sulu quickly realised that the only way back to his universe was via the Janus Vortex that was now in Tholian space. To get there, the Excelsior had to contend with the mirror versions of Klingons, Romulans and Tholians, plus the ISS Enterprise-A under the command of Pavel Chekov, who had become increasingly suspicious of his comrade.
After a difficult journey, the Excelsior managed to switch back places with its mirror universe counterpart, that had been easily contained by Commander Pavel Chekov in command of the USS Enterprise-A. (TOS video game: Shattered Universe)
The Vortex existed in two universes simultaneously. In the standard universe, it existed within Federation space and was located at Janus Prime. Within the mirror universe, it was located within Tholian territory and was known as Janus Ultima.