||We could use some "summerizing" of our aspect here: it's the dead of winter in PA ;-).
Multiplying the aspect grid values by a constant factor cannot possibly help. The problem is that aspect is not well-defined (aspects are defined only up to multiples of 360). Consider, for instance, the case where some aspects are 359 and others are 1. No matter how you slice it, the average will come out wrong. Dan, your approach works only when averaging two values at a time and it gives wrong answers in most cases anyway (because you have doubled all the angles, of course).
The correct solution depends on how you will use the averages, so I cannot recommend one in particular. But one method that will give reasonable answers is to convert each aspect into a (2D) vector, average them, and convert back to an aspect. This is easy: create cosine and sine grids, average them by zone, then recombine the results in the Field Calculator using an appropriately coded version of the arctangent (of which many examples abound in this forum: just search).
Other approaches are variants of this one. In many cases, you will also need the slope grid along with the aspect grid. From these you would create 3D vectors (the surface's unit normal field), average these, project them back into the coordinate plane, and compute an arctangent. It's the same as the first approach except you are computing a weighted average with the weights given by the cosine of the slope. This might be the appropriate mean value to use for insolation analyses, for instance, or for grid generalization.