TOKYO, July 8 (Reuters) - The euro slipped on Monday after projections from France's election pointed to a hung parliament amid an unexpectedly strong showing for a left-wing alliance, spawning fresh uncertainty over the country's fiscal outlook.

The dollar remained on the back foot following surprisingly soft U.S. payrolls data on Friday, which boosted bets for the Federal Reserve to soon start cutting interest rates.

Sterling rose to a 3-1/2-week top versus its U.S. peer as the British currency continued to firm following the Labour Party's landslide election victory last week, which ended 14 years of Conservative rule.

The Australian dollar benefited from greenback weakness to push to a six-month peak.

The yen headed for a third day of gains after rebounding from last week's nearly 38-year trough to the dollar.

The euro was 0.1% lower at $1.08235, and earlier slid as much as 0.4%, as investors weighed the consequences of a hung French parliament.

That was among several surprises in projected results, including the likely first-place finish for the leftist New Popular Front (NFP) alliance, and last-place showing for Marine Le Pen's nationalist, eurosceptic National Rally (RN), which had been the frontrunner going into Sunday's vote.

Polling agencies forecast the left would get 184 to 198 seats - well short of the 289 seats needed for an absolute majority. President Emmanuel Macron's centrist alliance was expected to get 160 to 169 seats, and the RN and its allies 135 to 143 seats.

"The shock of a far-right party gaining a majority was avoided in the short-term, but the rise of left-wing parties and increasing political uncertainties could raise concerns over French fiscal deficits and euro zone stability in the medium term," said Ken Cheung, director of FX strategy at Mizuho Securities.

The dollar index, which measures the U.S. currency against the euro, sterling, yen and three other major rivals, was flat at 104.95, licking its wounds after a nearly 1% slump last week, exacerbated by Friday's softer U.S. jobs market reading.

Traders currently set about 76% odds for a rate cut at the Fed's September meeting, up from 64% a week ago, according to the CME Group's FedWatch Tool. A subsequent cut is predicted by December.

The dollar slipped 0.14% to 160.615 yen, down from as high as 161.96 on Wednesday.

Sterling edged back 0.03% to $1.28105, after earlier rising to $1.2820 for the first time since June 12.

The Aussie was flat at $0.67465 after earlier edging as high as $0.67615 for the first time since Jan. 4, as the risks of another central bank rate hike contrasted with increasing bets on a near-term cut by the U.S. Fed.

(Reporting by Kevin Buckland; Editing by Jamie Freed and Sherry Jacob-Phillips)