BEIJING, April 3 (Reuters) - China's services sector struggled to get back on its feet in March after a brutal month of unprecedented shop closures and public lockdowns amid the coronavirus outbreak, a private survey showed on Friday.

Services companies cut jobs at the fastest pace on record as orders plunged for the second straight month and businesses scrambled to reduce their operating costs. Export orders also slumped again as more countries imposed their own tough virus containment measures.

While the Caixin/Markit services Purchasing Managers' Index (PMI) rebounded to 43 in March from a record low of 26.5 in February, it still remained deep in contraction territory and was the second weakest reading since the survey began in late 2005. The 50-point mark separates growth from contraction on a monthly basis.

The findings add to fears that consumer-facing services firms could be hit much harder and longer by the downturn than factories, which are slowly getting back to work, albeit at below normal levels. Similar business surveys will be released for Europe and the United States later on Friday.

The services sector is an important generator of jobs in China and accounts for about 60% of its economy, which now looks likely to shrink for the first time in 30 years. Many companies are smaller, privately owned firms with much less cash to see them through an extended slumps than larger, state-owned enterprises.

While some restaurants, malls, and movie theatres are slowly reopening as China eases restrictions, many remained closed or are operating at limited capacity as authorities sound the alarm about a possible second wave of infections and advise people to avoid gatherings.

Schools remain shut in most part of the country, and some cities and regions, including a county in China's central province of Henan, have introduced strict quarantine measures in light of new infections.

"Services activity remained under huge pressure and continued to shrink markedly amid restrictions to contain the coronavirus epidemic," Zhengsheng Zhong, director of macroeconomic analysis at CEBM Group, wrote in a note accompanying the Caixin PMI release.

"The recovery of economic activity remained limited in March, although the domestic epidemic was contained," he said.

The slump in the private sector survey, which focuses more on small, export-oriented companies, also contrasted with an official survey this week, which showed an expansion in activity.

The rapid spread of the pandemic outside China has put more pressure on demand, CEBM Group’s Zhong said, and wreaked havoc on global supply chains.