Upgrading from Exchange 2003 to Exchange 2010, routing works in one direction only
Solution 1:
I would run Get-RoutingGroupConnector
and verify that you have a connector from your EX03 back to your EX10 environment, and from your EX10 to your EX03. Although, your test seems to indicate that you DO have a connector from your EX10 to your EX03. You may just be missing one from your EX03 to EX10.
IF so you can create a new connector using
New-RoutingGroupConnector -Name ex10-to-ex03 -sourcetransportserver ex03server -targettransportserver ex10server
Add -PublicFolderReferralsEnabled $True
to the end of the above command if you need access to Public Folders, across the connector.