I am pretty ignorant about complex analysis so please forgive my lack of terminology.
I saw a pretty picture (posted below) of the behavior of the Riemann zeta function along the critical line. What I noticed is that the function takes on negative imaginary values "just before" the zeros and positive imaginary values "just after" them. I would describe these as having the same "positive orientation" in that sense.
(EDIT: Actually not in that sense; if the zeros were on the other side of the loops they would go from positive to negative but still have the same "orientation" inutitively. So I'm not as sure as I thought I was that I could formalize this)
A couple of obvious questions spring to mind:
1) Does this keep happening? The behavior going on at the beginning shows one way in which it might deviate from a spiral shape, where it seems to twist itself into the opposite orientation.
2) Is this interesting? Obviously this can only tell us anything about zeros on the critical line so it's not saying anything interesting for the big problem :) But a "yes" answer to this question might be something like: there is a more useful notion of "orientation of a zero" which looks at a whole neighborhood of the zero. Or: there are function where a similar restriction and analysis tells us something interesting about some structure the function possesses.