OK, well, I'm having trouble visualizing this.
Let me use my sweater as an example. Here is what I see. You would start w/ a block of white, a block of orange, another block of white, and then another block of orange.
When I start the block of white, I'm going to end up w/ the white working yarn moving from the left to the right ... aha...
.... I just visualized why this won't work. Every time you knit a row of a color, you end up w/ the working yarn on the right "end" of the work, and the next time you come around, you don't have a working yarn to knit with. Now I get it.
So, I guess this probably precludes this particular project from being knit in the round, but I'm not giving up yet. I think I'm going to have to pull out some needles & fiddle with this. I have an idea.