By Gertrude Chavez-Dreyfuss NEW YORK, Oct 26 (Reuters) - U.S. Treasury yields were mixed on Tuesday in thin volume, with those on the long end of the curve falling for a third straight session as investors looked to next week's Federal Reserve meeting for clues as to the timing of its first interest rate hike in three years. U.S. 2-year debt prices were lower on the day ahead of the note's $60 billion auction. Traders tend to sell Treasuries before an auction so they can buy them back at a lower price in a move called "supply concession." "Little has changed for interest rate fundamentals since last Thursday, despite slightly higher bond prices," wrote Jim Vogel, senior rates strategist, at FHN Financial in a research note early on Tuesday. "Actually, inflation expectations continue to increase this morning. Now that markets see central banks altering their policy guidance to consider less easing in the face of supply-demand induced higher prices, they have to grapple with the much harder question of where short-term rates need to be to slow rising prices," he added. The U.S. 5-year inflation breakeven rate, which reflects inflation expectations over the next five years, hit another milestone on Tuesday, rising to 2.97%, the highest since at least January 2004. Fed funds futures are still fully pricing a rate hike in September next year, and another increase in December. In morning U.S. trading, the benchmark U.S. 10-year yield was down at 1.6308%. The U.S. 5-year yield, which reflects Fed tightening, was up less than basis point 1.1844%. It touched an eight-month high of 1.193% last week. U.S. 30-year yields were also down, nearly 2 basis points at 2.0690%. Ahead of the auction, the U.S. 2-year yield was up 1 basis point at 0.4519%. FHN's Vogel noted that volume was thin for a third straight session, adding that the longer this runs, "the direction of the next sizeable move will grow in technical importance." The U.S. yield curve, meanwhile, flattened on Tuesday, which reflects a bet on a Fed rate hike sooner than later. The spread between U.S. 5-year notes and 30-year bonds narrowed to 87.8 basis points. October 26 Tuesday 9:07AM New York / 1307 GMT Price Current Net Yield % Change (bps) Three-month bills 0.055 0.0558 0.000 Six-month bills 0.0625 0.0634 0.000 Two-year note 99-158/256 0.4498 0.013 Three-year note 99-150/256 0.7664 0.013 Five-year note 98-134/256 1.1844 0.006 Seven-year note 98-144/256 1.469 0.002 10-year note 96-152/256 1.6273 -0.008 20-year bond 95-60/256 2.0438 -0.019 30-year bond 98-148/256 2.0641 -0.021 DOLLAR SWAP SPREADS Last (bps) Net Change (bps) U.S. 2-year dollar swap 17.50 0.25 spread U.S. 3-year dollar swap 16.00 0.00 spread U.S. 5-year dollar swap 8.75 0.00 spread U.S. 10-year dollar swap 3.00 0.25 spread U.S. 30-year dollar swap -19.25 1.00 spread (Reporting by Gertrude Chavez-Dreyfuss; Editing by Andrea Ricci)