For several years, the Justice League of Earth-One and Justice Society of Earth-Two traveled to each other's world under special circumstances usually during a crisis of one type of another. While each Flash mastered the ability to bridge the gap that separated their two planes of existance, it proved difficult most times of the year. As their nutual friendship developed over time, the desire arose to develop the means to freely traverse the Multiverse in order to meet regularly on more social terms.
Hence, the two groups jointly devised technology allowing each team to visit each other's planet on any occasion, not just for their annual gatherings but if special needs arose. Utilizing physics similar to the teleportation tubes the League also employed to transport its members from planetside to their orbiting satellite headquarters, on occasion the cubes were known to malfunction. In fact, the first time they were used, the android Red Tornado secretly snuck into the chamber while three of his new League allies were in mid-transport, exchanging places with three Socialites who were doing the same. Due to Tornado's desire to his native Earth-Two, his extra mass caused the three-on-three dimensional swap to be out of balance, leading to both teams being sent to Earth-X where the seven heroes assisted that world's Freedom Fighters.
Later, Professor Potter accidently activated the Transmatter Cube on board the JLA satellite while Earth-Two's Superman was travelling to Earth-One to enlist his counterpart's aid in saving the life of an elderly Jimmy Olsen. Instead, the beast-man known as Krogg was unleashed, leading to the Men of Steel merging their bodies into a "Phoenix of Steel". Shortly thereafter, despite the protests of Hawkman, Green Arrow and Black Canary used the device to travel back to the Pretty Bird's native Earth-Two, which caused a divergent Earth-One reality to form that a quintet of heroes had to restore to normal. And on yet another occasion, due to the machinations of Per Degaton, a divergent reality on Earth-Two was created due in part to the Cube's bringing the Crime Syndicate into the fray with their heroic dopplegangers.
Still, when not wrecking timelines and creating its own handy plot device, the Cubes were a useful gadget for the Greatest Heroes of Two Worlds!