Would it be possible for the sun to rise in the west in the spring, and as the year progress, move slowly around the compass, day by day. So one day it’ll rise in the west, the next a tiny bit south…
Would this work?
This is just off the top of my head, without any calculations, but here goes:
The only way I could see this working is if you were standing at a pole of a planet that rotates very slowly (once per ‘year’) orbitting its star very quickly (once per ‘day’). The planets axis would have to lie on the plane of its orbit. Of course, this plays very fast and loose with the meanings of the words ‘year’ and ‘day’ as well as ‘east’ and ‘west’.
I suppose you could hypothesize a planet of very large mass that is the center of its system. It would have a small star that orbitted it. Not sure how you could have something that big without it being a star itself, at some point. Maybe a burned out brown dwarf?
Again, I’m not an astronomer or a cosmologist, so I’m just tossing this out off the tip of my gray matter.
Jeeze i wish i was smart.
Y’know Satire I think that just might work.
For example, if the planet rotated (i.e. turned about its axis) 4 times every time it revolved (i.e. made its trip around the sun), the first day* could begin with the sun rising in front of you and setting behind you, the second day* would have the sun rising to your left and setting on your right, and so on for a four day cycle until it rises in front and sets behind you again (note, this was simulated by rotating my left hand around my right fist and slowly twisting it. My roommate was giving me rather odd looks).
- the problem here, unfortunately is that “day,” or a time in which the sun rises and sets, occurs not due to the rotation of the planet (as it does on Earth), but due to the revolution of the planet about its star. Interesting way to thing about it. And it essentially amounts to the star rotating about the planet (as Satire’s second solution mentioned)
Two words: Binary stars.
I really doubt it’s plausible, but perhaps the planet is in a distant orbit from one star, but close enough to the second star (the star which it doesn’t orbit) to receive light/energy. I’m trying to envision the exact model for this system, but it’s 3:00 AM and my mental faculties aren’t working right, but perhaps you can play around with that idea.
How about 3 or 4 stars of the same mass equidistant in orbit around the planet? Since the planet is in the exact centerpoint of their orbit it doesn’t matter what mass it is. 
God, I’d hate to imagine what fucked up forces of physics would cause such a system to form…