Unless I am misreading, I believe your statement is incorrect. Annual dues payments have nothing to do with use year, but rather calendar year. Using your example, if you have a June UY and add on in May 2009, you will start paying annual dues for that add on in May, if you have it set up monthly, or pay the whole May-Dec 2009 dues at once. 2010 dues will be due in January 2010.
Someone feel free to correct me if I am wrong.