Here is a hack:

• Read the color from object's vertex colors, this way your object's can share the same material with different colors:

  

• Change the vertex colors on frame change with python script. This won't have real-time performance but that shouldn't matter since you don't get material preview with vertex-colors in view-port anyway (it shows only in render).

   import bpy
  
   # this is how you add vertex-color map:
   obj.data.vertex_colors.new(name='Col')
  
   # this handler function, it will be called on every frame change
   def update_colors_from_rotation(scn):
       for obj in scn.objects:
           color = YOUR_FUNCTION(obj.rotation_euler)
           # color should be float array [R,G,B]
           # or you can access custom ID props here also: color = obj['base_color']
           # and do something with it
  
           # here is how you change the vertex-color map:
           for ipoly in range(len(obj.data.polygons)):
               for ivertex in obj.data.polygons[ipoly].loop_indices:
                   obj.data.vertex_colors['Col'].data[ivertex].color.r = color[0]
                   obj.data.vertex_colors['Col'].data[ivertex].color.g = color[1]
                   obj.data.vertex_colors['Col'].data[ivertex].color.b = color[2]
  
   # register the function in frame change handlers:
   bpy.app.handlers.frame_change_post.append(update_colors_from_rotation)
  

Here is a hack:

• Read the color from object's vertex colors, this way your objects can share the same material with different colors:

  

• Change the vertex colors on frame change with python script. This won't have real-time performance but that shouldn't matter since you don't get material preview with vertex-colors in view-port anyway (it shows only in render).

   import bpy
  
   # this is how you add vertex-color map:
   obj.data.vertex_colors.new(name='Col')
  
   # this handler function, it will be called on every frame change
   def update_colors_from_rotation(scn):
       for obj in scn.objects:
           color = YOUR_FUNCTION(obj.rotation_euler)
           # color should be float array [R,G,B]
           # or you can access custom ID props here also: color = obj['base_color']
           # and do something with it
  
           # here is how you change the vertex-color map:
           for ipoly in range(len(obj.data.polygons)):
               for ivertex in obj.data.polygons[ipoly].loop_indices:
                   obj.data.vertex_colors['Col'].data[ivertex].color.r = color[0]
                   obj.data.vertex_colors['Col'].data[ivertex].color.g = color[1]
                   obj.data.vertex_colors['Col'].data[ivertex].color.b = color[2]
  
   # register the function in frame change handlers:
   bpy.app.handlers.frame_change_post.append(update_colors_from_rotation)