* Euro falls after Italy, euro zone figures disappoint
* Dollar index holding below 10-1/2 month peak
* Swedish crown, sterling jump after high PMI readings
* Aussie pares losses after dip on China HSBC services PMI (Recasts after euro zone data)
LONDON, Aug 5 (Reuters) - The euro headed back below $1.34 on Tuesday after data on Italy's service sector proved weaker than expected, while a surprise bounce for equivalent numbers in Sweden and Britain drove the crown and the pound higher.
The European PMI surveys as expected all showed growth, but the dip in the Italian number underlined a fragile outlook that has the market expecting the European Central Bank to pump yet more euros into the financial system.
Overall numbers for the euro zone were also revised down a touch from the initial flash estimate last month.
"The euro certainly has reacted negatively to the Italian numbers," said Neil Mellor, a strategist with Bank of New York Mellon in London.
"On the main euro zone numbers even 54 is not exactly a figure that sets the world alight. It returns us to what we've been seeing for a few weeks now - that the momentum in Europe is faltering."
The data pushed the euro, which is down as much as 6 cents against the dollar since early May, almost 0.2 percent lower in morning trade in Europe to $1.3395.
As yet, that did not look like the tiebreaker markets have been waiting for on the dollar's strength.
Speculation that the greenback could be beginning a longer-term push higher have grown after its best performance in a year and a half in July.
But a slightly softer than expected U.S. jobs number at the end of last week allowed some to take profits on that progress, leaving the dollar stuck just below a 10-1/2 month peak against a basket of currencies.
"We see upside risks for the dollar today, given our forecast for a solid July ISM Non-Manufacturing," analysts from Dutch bank ING said in a morning note, referring to U.S. data due later on Tuesday.
"This should reaffirm the view that the U.S. economy is on the right track and the July U.S. labour market report was not a game changer. Thereafter, (we expect) the dollar index to stay range-bound for the rest of week."
Sweden has been symbolic of Europe's flirtation with very low inflation, or even falling prices, this year, but its services PMI for July shocked by surging to 60.1 points from 54.6 a month earlier. That took the crown a third of a percent higher to 9.2042 crowns per euro, from 9.235 beforehand.
"It had been trending higher but this is a surprise," Citigroup strategist Josh O'Byrne said, adding that he viewed the crown as fairly valued at current levels against the euro and dollar.
"The momentum has been good in Sweden in recent months and if inflation does begin to pick up as the Riksbank expects then we may get back to seeing some upward potential later this year," he said.
Sterling has also suffered in the past week, falling almost 1 percent against the greenback, but Britain's economic outlook is more positive than that for the euro zone.
The purchasing managers' index (PMI) for the services industry, which makes up around 78 percent of the British economy and employs about 80 percent of its workforce, rose to an eight-month high of 59.1, beating a forecast for 57.9.
"The survey signalled that economic growth in the UK continues to remain robust," said Lee Hardman, a currency economist at the Bank of Tokyo-Mitsubishi UFJ in London, who expects the BoE to hike rates in November.
"Expectations had been building that the economy is likely to lose momentum later this year, but today's survey has dampened those more dovish expectations somewhat, so that's helping to give the pound a lift in the near term."
Sterling hit a day's high of $1.6890 after the data from $1.6854 beforehand. It was last trading at $1.6879, up 0.15 percent on the day. (Additional reporting by Ian Chua in Sydney and Masayuki Kitano in Singapore; Editing by Louise Ireland/Ruth Pitchford)