* Private index on euro zone business hits 10-month high * U.S. services sector grew at slower pace in January -ISM * Fed to buy $1.25 bln to $1.75 bln debt due 2036-2042 * U.S. sells record amount of one-month bills * Long-dated yields rise on talk of "exotics" hedging By Richard Leong NEW YORK, Feb 5 (Reuters) - U.S. Treasuries prices fell on Tuesday as a rebound in Wall Street stocks and less gloomy data on European business activity cut the appetite for safe-haven government debt, pushing benchmark yields back above 2 percent. Adding to the bond selloff, yields on Italian and Spanish sovereign debt fell, after a jump on Monday on worries about possible political shake-ups in the two countries, the euro zone's third and fourth biggest economies. Selling of long-dated hedges on debt securities that are known as power reverse dual currency notes also weighed on Treasuries prices. Dealers have been unwinding hedges on these "exotic" products due to weakness in the Japanese yen against the dollar, traders and analysts said. "The key in the near term is the movement in Spanish and Italian yields. We are seeing a small reversal after Monday's move. We are seeing Treasuries yields rising in response," said Brian Reynolds, chief market strategist at Rosenblatt Securities in New York. Benchmark 10-year Treasuries notes were 17/32 lower in price at 96-17/32, yielding 2.018 percent, up 6.1 basis points from late on Monday when the 10-year yield hit a more than nine-month high. The 30-year bond was 31/32 lower in price at 91-8/32 for a yield of 3.208 percent, up 4.6 basis points from Monday's close. On Wall Street, the three major stock indexes rose after falling on Monday in the biggest one-day drop since November. "It's a reversal in risk appetite after yesterday," said Richard Gilhooly, fixed-income strategist at TD Securities in New York. Risk appetite was helped by a partial recovery in Italian and Spanish government debt, a day after their investors sold on worries of political instability in Spain and Italy. The yield on 10-year Spanish government notes fell from Monday's six-week high to 5.37 percent, down 6 basis points, while the yield on Italian sovereign debt slipped to 4.45 percent, down 2 basis points from late on Monday. There were also signs the euro zone's economy may be turning the corner. Markit's Eurozone Composite PMI, which is based on business activity across thousands of companies, rose in January to a 10-month high of 48.6 from 47.2 in December - an improvement on the preliminary reading, though still below the 50 mark that divides growth and contraction. The Institute for Supply Management said U.S. services industries grew at a slower pace in January. The data was consistent with a moderately growing economy and allayed fears that the slight contraction in U.S. growth in the fourth quarter might turn into something more lasting. On the supply front, the U.S. Federal Reserve will buy $1.534 billion in Treasuries that mature between February 2036 and November 2042, which is part of its $44 billion purchase of Treasuries in February. The U.S. central bank has been buying Treasuries and mortgage bonds in an attempt to lower borrowing costs and stubbornly high unemployment. The U.S. Treasury Department sold $45 billion in one-month bills, which is a record amount for this maturity. These one-month T-bills fetched an interest rate of 0.065 percent, compared with 0.035 percent on the $30 billion of one-month bills sold last week. EXOTICS MOVE In the derivatives market, there were signs of reduced demand for long-dated, fixed-rate payments in the dollar interest rate swap market, which some analysts said was due to dealers unwinding hedges on power reverse dual currency notes. Some Japanese investors have held these "exotic" but risky debt products, on which they were receiving higher coupon payments when the yen was rising against the dollar in the aftermath of the global financial crisis. Since last fall, speculation the Bank of Japan would ramp up monetary stimulus to help the economy led to a dramatic weakening of the yen versus the dollar. On Tuesday, the yen hit a 2-1/2-year low against the greenback at 93.51 yen, down about 21 percent from September, when speculation on the BoJ started. With the yen weakening, the coupon payments on these notes should fall, and dealers who sold these notes will likely need less long-dated cash payments to ensure they meet their coupon payments to investors, analysts said. The discount on the 30-year dollar swap rate to the 30-year Treasury bond yield, which narrows with lower demand for long-dated, fixed-rate cash flows, briefly narrowed to 15.25 basis points, a level not seen since March 2010. It widened out to 16.25 basis points, unchanged on the day.