While fishing, I noticed that when I was sneaking on the ice between an ocean and a snowy beach biome, my movements seemed to be unusual somehow. Through experimentation I discovered that while holding the sneak key, if I alternated movements between two opposite directions (eg, left and right or forwards and backwards) while not changing the direction I was facing, my direction of movement would change even though I was still facing the same way. It seems that after the first move (eg forwards in a sequence of forwards, backwards, forwards, backwards etc.), the direction of movement would change to line up with the x or z axis even though I was facing diagonally across those axis so I should have moved diagonally. (note: fishing is not a part of this glitch, it's just what I was doing when I discovered it)
So basically, when you take a few steps from side to side or forwards and backwards while sneaking on ice, your direction of movement can change by up to 45 degrees.
Once you have achieved the effect, it will persist for as long as you hold crouch and remain on the ice, and you do not need to continue alternating directions. This makes it easier to see the effect, because you can just go in one direction and observe how your direction of travel does not correspond to the direction you are facing. Doing this and moving onto another type of block (not ice) will end the effect, making it even more obvious because you will now move in the correct direction.
Achieving the effect to begin with is difficult;
All of these effects are observable while pressing only a single movement direction at a time.
I tested this in creative, and it worked. Using the test shown in the attached picture, it is possible to achieve the effect on the ice square under the arrow (you don't need much room). You can then use it to sneak down the line between the pressure plates while facing diagonally, without touching any of the pressure plates. This proves that it is possible to move along that axis despite not facing along it.