(coexisting on another dimensional plane)
Interdimensional contact was the term used to describe the passage between universes, namely the prime and mirror universes. (TOS: "Mirror, Mirror") In the mirror universe, this action was referred to as a crossover. (DS9: "Crossover") While in another context, Beckett Mariner, of the prime universe, informally described the event as the "mirror universe switcheroo". (LD: "We'll Always Have Tom Paris")
Following his arrival in the mirror universe, Captain James T. Kirk hypothesized that the ion storm they beamed through was of "such magnitude" that it caused a power surge in the transporter circuits and created a momentary interdimensional contact with a parallel universe, at which moment, those in each universe were transposed with their counterpart in the other universe. (TOS: "Mirror, Mirror") This is indeed how, a decade earlier, the Gabriel Lorcas of the Prime and mirror universes switched places with each other, though this information was ultimately classified by the Federation Starfleet due to the ongoing Federation-Klingon War. (DIS: "What's Past Is Prologue", "The War Without, The War Within")
According to Intendant Kira Nerys, "After the first crossover, we were afraid that others might come to interfere in our affairs. It was decided then that if it ever happened again, we would promptly dispose of anyone who appeared from your side." Later, Kira explained that because the first crossover involved a transporter, "all transporters were redesigned afterwards to make sure the same thing would never happen again." (DS9: "Crossover")
Years later, when Quark and Rom were imprisoned in the mirror universe, Quark attempted to demean the Elim Garak of that universe by stating that "[i]f you were smart, you'd crossover to our universe and get some pointers from our Garak. When it comes to torture, assassination and sabotage, you could learn a lot from him." (DS9: "The Emperor's New Cloak")
Following Shaxs' unexplained resurrection in 2381, Mariner considered such an event to describe his mysterious return. (LD: "We'll Always Have Tom Paris")