Solution to problem in three easy steps.
1) Using original regular reference grid, use 'Feature to Point' tool with 'Inside' option checked to get centroid of grid cell.
2) Using 'Editor', move your points to the relevant corner of your grid.
3) Using the ESRI IGL Font22 a number of triangles are available either to use as is, or to rotate accordingly (although all four corner orientations are available) to symbolise data.
You will need to fiddle with the positioning of points and sizing of triangles to make data fit accordingly.
I found that by moving centroids by 90% relative to distance from centre to either edge and setting symbol to size 20 managed to get the desired result.
Hope this helps and reduces amount of work.
Regards,
Craig