* Euro zone factory slowdown deepens in December * China's economy showing signs of revival * U.S. PMI shows modest growth, export orders rebound By Jonathan Cable and Lucia Mutikani WASHINGTON/LONDON Jan 2 (Reuters) - Manufacturing in the United States and China grew in December, suggesting the global economy was on course for moderate growth this year, even as the euro zone looked set to sink deeper into recession. U.S. factory activity rebounded last month after stumbling to a 40-month low in November, with new export orders growing for the first time since May. That mirrored growth in Chinese manufacturing reported earlier this week. The Institute for Supply Management said its index of U.S. manufacturing activity rose to 50.7 from 49.5 in November. The improvement came despite concerns over a wave of sharp government spending cuts and high taxes, the so-called fiscal cliff, that would have sucked $600 billion from the U.S. economy and push it into recession. In much of Europe, the mood was downbeat. Purchasing managers' surveys in the 17-nation euro zone showed economic decline among some of its biggest member countries. Markit's Eurozone Manufacturing Purchasing Managers' Index (PMI) edged down to 46.1 in December from November's 46.2, below a preliminary reading of 46.3. It has been below the 50 mark that divides growth from contraction since August 2011. "The reports are consistent with an expanding global economy, but one which is doing so, at least on the manufacturing side, at a relatively slower pace," said Jay Bryson, global economist at Wells Fargo Securities in Charlotte, North Carolina. In China, a manufacturing survey by HSBC, which focuses more on smaller and mid-sized firms, suggested activity was at its strongest since May 2011. China's official manufacturing PMI, released on Tuesday held steady in December at 50.6, matching November's seven-month high. The surveys add to signs economic activity in China picked up between October and December after growth slowed for seven consecutive quarters to 7.4 percent in the third quarter. That helped offset persistent weakness in Europe and Japan. In the United States, the rebound in export orders last month offered hope that trade would soften some of the blow on the economy from tighter fiscal policy. While Congress and the Obama administration managed to steer the economy away from the worst of the so-called fiscal cliff, Americans face a higher tax burden this year and government spending cuts. Analysts estimate that could cut as much as one percentage point from gross domestic product this year. "The rise in export orders is an indication that global economic momentum may also have improved this month," said Millan Mulraine, a senior economist at TD Securities in New York. DEEPENING SLUMP IN EUROPE Germany, Europe's largest economy, saw its crucial manufacturing sector shrink for the 10th straight month and at a faster pace, while French data showed a decline in all but one of the past 17 months. The slump in Spain deepened and Italy's index, although improved, remained below 50 for the 17th month. Ireland was the only member of the currency union to show manufacturing growth in December. "It's pretty grim really," said Jonathan Loynes at Capital Economics. "These surveys are pointing to a pretty deep recession. If the German industrial sector is contracting quite sharply, it is pretty hard to see where growth across the euro zone as a whole is going to come from." Economists are cautiously optimistic the euro zone economy will start growing at least by the second quarter of this year. British factory activity jumped unexpectedly to its fastest pace since September 2011, raising the chance that the economy eked out some growth at the end of 2012. Manufacturing activity expanded in Asia as a whole, driven by revival in China's economy, but export demand was uneven, pointing to further sluggish growth for the region. For Asia, much hinges on the pace and quality of the recovery in China as a new generation of leaders prepares to take charge. In India, Asia's third-largest economy, the HSBC Markit Manufacturing PMI, which gauges the business activity of the country's factories, but not its utilities, jumped to 54.7 in December from 53.7 in November, its biggest monthly rise since January 2012. Activity in Southeast Asia's largest economy, Indonesia, expanded but at a slower rate, as growth of new export orders eased from a month earlier. A lot may depend on elsewhere, however. "Asia is gradually improving, but the region, including China, remains largely exposed to exports and without signs of improvement in the U.S. and Europe it will be hard for activity to take off," said Frederic Neumann, co-head of Asian economics at HSBC.