Warning! Spoilers ahead for Batman/Catwoman #1 from Tom King and Clay Mann
In the new Batman/Catwoman maxi-series from DC Comics, writer Tom King revisits his 2016 run on Batman with this special 12-issue conclusion featuring the Bat and the Cat's romance. However, The Joker is also present in this first issue, and his own relationship to Catwoman is explored further, specifically in relation to their shared infatuation with Batman himself. As such, there is a sort of intermingling reminiscent of a love triangle that King forms between the three characters, though not in the traditional sense (this is Joker, Batman, and Catwoman after all).
In Batman/Catwoman #1, the series features three different interconnected narratives taking place at different times. The first takes place during the very beginning of Batman and Catwoman's relationship, the second sees the pair looking for the son of Andrea Beaumont (The Phantasm), while the third features an older Selina visiting a trailer park in Florida, having lived a long life with Bruce and their daughter Helena becoming Batwoman. While all three stories feature the Joker at some capacity, the first narrative features the most intriguing elements.
As referenced above, the earliest narrative takes place at the beginning of the Bat and Cat's romance, and Joker himself has some opinions about it, which he shares with Selina in a casual rooftop conversation the pair once had. The entire exchange is quite casual, holding the air of a conversation between fellow colleagues. However, there does seem to be some flirtation as well. Nothing to infer that anything romantic is going on between the two, but enough that the familiarity shared is striking. At the very least, it seems Catwoman is brave and confident enough to not fear Joker as others do, and the Clown Prince of Crime perhaps respects her for that as well as her personality and abilities, seeing her as "fun". However, he doesn't think she should entangle herself with the Dark Knight, the man they're both obsessed with (albeit in different ways).
It should be noted that this conversation has to have occurred before Batman and Selina's engagement to be married in King's original run. It's then that Batman fans remember that Joker and Catwoman fought each other, as Joker was determined to convince Selina that she shouldn't marry Bruce, as the happiness they'd share would ruin who he is as Batman. However, even then the kinship between the two rogues is felt as they fight.
The relationship dynamic between Catwoman and Joker in relation to Batman is one that King has long been fostering, and it seems as though he will be continuing to shine a light on it as Batman/Catwoman continues. Even in the future timeline, the older Selina is revealed to be meeting with an elderly and retired Joker. Initially, the pair had been reminiscing about their children and the good old days, when Selina shares that Bruce had died, providing the reasoning for her visit. Now that Batman is gone, he can no longer stop her from killing Joker for what he did to Andrea Beaumont in the present timeline (which is when the old man is revealed to be Joker). However, despite Catwoman's intent to kill, the familiarity and veil of friendship still persists. It's truly compelling and one of the most interesting elements of the new maxi-series thus far.