see here http://resources.arcgis.com/en/help/main/10.1/0154/01540000030s000000.htm
abrown31,
Did you find the answer to which outboard ports does ArcGIS Desktop use to contact ArcGIS Server? We are in a similar situation needing to cross multiple networks (DMZs). We can connect to ArcGIS Server using ArcGIS Desktop, but cannot publish to it. Error message says "unable to establish connection to the server." My PC cannot ping the ArcGIS Server.