From watching Evangelion, it becomes apparent to me that Rei displays
personality traits and appearance characteristic of both the goth and punk
movements.

Her passive-aggressive demeanor, the fashionable apathy she shows towards
others and her obsession with death (specifically her own) remind me very
strongly of the 'teen goth ethos'. Her pale white skin and red eyes also
remind me of the white makeup and 'demonic' contact lenses commonly worn by
goths in clubs (or at least the ones I've been to).

OTOH, Rei's 'I don't care what you think of me' attitude and desire to march
to the beat of her own drum is pure punk attitude, IMO. Her idiosyncratic
dress-sense (school uniform is no longer about conforming to the standard
and destruction of individuality when you wear it ALL the time - it then
becomes a statement), frank sexuality and wild, brightly-colored, slightly
androgynous hair somewhat support this assertion.

I'm not sure if anyone will agree with me here, but I find myself wondering
if Anno/GAINAX purposefully intended this for Rei's character in the
beginning - before the whole Yui/cloning/soul of Lilith thing was even
thought of.

I'd be interested to hear your opinions on this matter, especially those of
any goths, punks or avid fans of Reichan who might be reading this.