Voting Integrity and Verification Act of 2005 - Amends the Help America Vote Act of 2002 to revise requirements for voting systems used in an election for Federal office, including to: (1) permit the voter to verify the accuracy of his or her ballot (in a private and independent manner) in an individual paper version before it is cast and counted; and (2) require the voting system to produce an individual permanent paper record for each ballot cast which meets specified requirements relating to manual audit capacity.