* Consumer confidence climbs for third consecutive month
* Industry outlook improves
* Unemployment rises in September
(Adds reaction, details)
By Vicky Buffery
PARIS, Oct 27 (Reuters) - French household confidence rose for the third straight month in October, official data showed on Tuesday, as consumers shrugged off accelerating job cuts in the euro zone's second largest economy.
Consumer confidence edged up to -35 in October, from -36 the previous month, national statistics office INSEE said, in line with a median forecast from a Reuters survey of economists.
Although the confidence indicator remained well below its long-term average, October's reading was the highest since January 2008.
"The surprise comes from the fact that households think the unemployment rate is going to increase less quickly, which is strange because the number of jobseekers increased again in September," said Mathilde Lemoine, chief economist at HSBC France.
Figures published on Monday showed job cuts accelerated in September, boosting headline unemployment by 21,600.
But this seemed to have little impact on October's consumer survey. Fewer respondents voiced fears about rising unemployment in the months ahead, while a larger proportion thought now was a good time to splash out on major purchases.
INSEE also released its quarterly industry survey, which showed firms expect demand to decline in the fourth quarter but at a markedly slower pace than in previous quarters.
Foreign demand improved over the third quarter of the year and is expected to remain stable between the months of October and December, the survey showed.
"These surveys together with the indication of the October PMIs seems to suggest that the recovery which is underway is stronger, more genuine than we could have thought," said Tullia Bucco, economist at Unicredit MIB in Milan.
The French economy grew by a surprise 0.3 percent in the second quarter of the year, but concerns have been raised that the recovery might run out of steam as rising unemployment cuts into household spending power.
"The French have clearly realised that the recession is over," said Asteres economist Nicolas Bouzou.
Economy Minister Christine Lagarde confirmed yesterday that she remained optimistic about the outlook for household spending, one of the main drivers of economic growth.
Coming in the wake of other upbeat indicators on the French economy, the data also paints an optimistic picture of the outlook for the fourth quarter and could help to allay fears that the nascent recovery might run out of steam.
"It is certainly encouraging for the fourth quarter gross domestic product outlook which probably won't soften much with respect to the third quarter," Bucco said.
October's flash composite purchasing managers' index released last week rose to its highest level in 35 months, reflecting a broad-based surge in business activity.
(Additional reporting by Anna Willard and James Mackenzie; Editing by Ruth Pitchford)