There is a the slope command in Spatial Analyst and you get the vertical angle with the horizontal plain using that. The aspect command gives you the 360 deg horizontal direction of the slope. The default symbology is the N, NE, E, SE, S, SW, W, NW directions, but the raster values are down to the sub degree precision. Maybe a combination of these two would work.
In Spatial Analyst hydrology the flow direction command gives you a number representing one of the 8 directions (N, NE, E, SE, S, SW, W, NW) that is unique to the four corners. This is slope is towards the lowest of the of the surrounding cells and may not be the same as aspect.
Hillshade produces a value based on horizontal direction, but I'm sure if it can be correlated to cardinal directions.
Just some ideas...