It would be possible, and not even terribly difficult, to have a faucet whose stream, viewed at a moment in time, would appear as a spiral. A double-spiral diamond pattern would be harder, but should by no means be impossible, though unless the diamonds were rather coarse they would tend to become blobby as water moved away from the faucet. The key observation, though, would be that all of the water in the stream would be moving in the same direction away from the faucet.
To understand what's going on, imagine that the pipe leading up to the end of the faucet was perfectly straight for the last six feet and one had a dowel in it. Imagine further that the end of faucet contained cutters around the perimeter which rotated as the dowel moved through. Even though the dowel wasn't rotating, the cutter would have no difficulty cutting a spiral groove in it (and a pair of counter-rotating cutter rings could cut a diamond double-spiral arrangement). Cutting neat-looking diamonds in water would be harder than with a solid dowel, but there are ways such a thing could be accomplished.
What's important to note, however, is that unless one is either taking a short-exposure photograph of the stream or illuminating it with a strobe light, it would appear as an ordinary stream of water. Stroboscopic illumination, however, could make the pattern appear to move slowly upward or downward along the stream (there would be a limit to how fast the pattern could appear to move and still be visible as a pattern). The strobe wouldn't change the rate at which the water actually moved, however--merely how the pattern within the water would appear to move.