Finally, years after the original debuted and his younger twin appeared, after several meetings between their teams...the Justice League and Justice Society of two Americas...the Supermen meet! The circumstances surrounding this encounter, however, did not bode well for their relationship.
After a titanic battle with the evil Aquarius, Superman and his JSA teammates saw the malevolent menace erase Earth-Two from their plane of existence. Left with the prospect of his adopted world forevermore lost as was his native Krypton-Two, Kal-L fell into deep despair with his surviving allies. As a result, they each became receptive to their foes influence, turning into crazier cronies of the cosmic creature. Hence, when the Justice League arrived in their universe, the Society vigorously battled their past colleagues. This included two Men of Might.
There have been varying explanations for how this result transpired. One such theory is that Earth-Two's Superman was empowered by Aquarius with energy from his purloined cosmic rod, stolen from Starman, and this holds some merit due to the nature of Earth-One's Superman. Since Kal-El was charged with yellow sun energy, perhaps similar stellar energy magnified the abilities of the Kal-L that were already genetically predisposed even on Krypton-Two. While naturally absorbing such energized particles would not in-and-of themselves elevate Kal-L's power levels, when concentrated artificially through such a mechanism would enhance his physique to temporarily match that of his cosmic twin.
Regardless of the science behind this standstill, the Kryptonians were equally matched in this instance. After the Society was freed from the influence of their foe, they joined the League in decisively defeating the demented deviant, causing Aquarius to self-destruct in a negative universe adjacent to Earth-Two (similar to the anti-matter universe adjacent to Earth-One) leaving the threat abated. As both teams part, there is no indication that the Men of Tomorrow have healed the breach caused by their adversary.
Subsequent meetings between the two would develop their flowering friendship in ways neither quite expected...