Perhaps the most desired teamup between counterparts that never occured was with the Batmen, who never actually met (although they did work on a case together tackling the same foe in different eras). However, a close second would be with the Last Daughters of Krypton, Supergirl of Earth-One and Power Girl of Earth-Two. Actually, they did meet once, albeit briefly.
When a cosmic entity known as the Arbitrator came first to Earth-One then three other parallel planes of existence including Earth-Two, Earth-X and Earth-I(mmortality) with his synthetic minions to plague those planets, Womder Woman enlisted the aid of heroines on her world and others. Among these were Kara Zor-El and Kara Zor-L, the former assisting Madame Xanadu and Phantom Lady on Earth-X and the latter assisting the Huntress and Black Canadry on Earth-Two. During their final confrontation with the Arbitrator, the Maidens of Might had a brief one panel interaction that showed a rather lack of mutual respect that their cousins shared. Then they parted along with their fellow heroines rather abruptly when this odd case concluded.
Although Power Girl returned to Earth-One for annual Justice League/Justice Society teamups, and Supergirl once joined the League on an adventure versus the Commander on Earth-Two... these two never again met. Perhaps the reason is the adjoining frame of a potential interaction between the two Girls of Steel. As a result, these Girls of Steel weren't able to build a lasting relationship like the Supermen, Flashes, Green Lanterns, and others of their two worlds. By the time they'd have a chance to start over with one another, the Crisis on Infinite Earths occured leading to the tragic death of Supergirl in the Anti-Matter universe, while Power Girl was left behind, saving her life.