AA is an premium grade that says the same veneer is used on both faces. It's real expensive and not all that available.
A1 is the best of the standard grade stuff, has the A veneer on the front and 1 grade (the best) for the back. Structurally it should not be a problem and ought to be pretty good appearance wide. A bigger issue is the water resistance. Usually you want something other than construction/cabinet plywood (which is represented by something other than the codes for face veneer grading).