Two famous characters which seem to share
certain parallels are Joker and the Cheshire Cat. It's not just because they
both wear unnatural, unsettling grins on their
faces. It's because they both represent order vs. chaos, but in different ways.
The Cheshire Cat acts as a guide among chaos,
providing advice and order at times--although he's capricious and only helps
when he feels like it. Rather than good or bad, he seems neutral and mainly
motivated by his own whims for amusement.
In the Batman universe, Joker is also motivated
by his own amusement and says perplexing things. But he wants to upend order
and create chaos, which is somewhat the opposite of Cheshire Cat. The latter
tries to provide balance between the two, whereas Joker just wants to destroy.
Also, both follow a motif of what they, in
their own words, call "madness." They're products of their
environments and arguably both well-adjusted to their surroundings, even if
others would see them as nonsensical or erratic. It reminds me of the aphorism
that it's no measure of health to be well-adjusted to a sick society. The Joker
has found a definite means of survival, living purely at the expense of others.
He's sustained himself, but not in a way that could ever be seen as healthy.
The Cheshire Cat seems better adapted because he accepts the inevitable
disorder but tempers it with organization, and he's far less malevolent.