As a matter of fact the dwg exported from ArcMap contains attribute data in form of Xrecords...
[url=http://www.cadenhancement.com/2009/05/xrecords-what-are-they.html]http://www.cadenhancement.com/2009/05/xrecords-what-are-they.html[/url]
...which have been a standard feature of dwg-files for a long time, but only accessible to developers.
ESRI offers a free plug-in for AutoCAD called ArcGIS for AutoCAD...
[url=http://www.esri.com/software/arcgis/arcgis-for-autocad]http://www.esri.com/software/arcgis/arcgis-for-autocad[/url]
...which allows you to use those Xrecords within AutoCAD to change your symbology. I don't use AutoCAD myself, so I can't give you any personal opinion about the performance of the plug-in.