(Adds U.S. market open, byline, dateline; previous LONDON)
* Grim PMI data in Europe
By Herbert Lash and Tom Arnold
NEW YORK/LONDON, May 6 (Reuters) - World equity markets were mixed on Wednesday, with U.S. indexes seesawing amid hopes for a pickup in business activity and as oil prices slumped with demand low after a rise in U.S. crude stockpiles last week to three-year highs.
The safe-haven Japanese yen and dollar rose after economic data showed U.S. private payrolls tumbled last month to a record 20.2 million, German industrial orders fell at a record pace in March and UK construction activity crashed to an all-time low in April.
The dollar index rose 0.359%, with the euro down 0.36% to $1.0799. The Japanese yen strengthened 0.49% versus the greenback at 106.10 per dollar.
Retail sales in the euro zone also suffered their largest decline on record in March.
Demand has collapsed globally because of the coronavirus epidemic and many investors viewed the prospect of a swift recovery as bleak, despite the gradual opening of economies worldwide that has given hope to other investors.
Trying to gauge economic growth in light of the pandemic has sent investors scurrying for new insights, especially in China, where the economy was first to re-open, said Yousef Abbasi, global market strategist at INTL FCStone Financial Inc in New York.
"When are you going to see those businesses retake capacity comparable to the last year? That's the big question," Abbasi said. "It really is about where you think demand is going to come back or where you think demand never really dropped off."
The pan-European STOXX 600 index lost 0.37% while MSCI's gauge of stocks across the globe shed just 0.07%.
On Wall Street, the Dow Jones Industrial Average fell 49.71 points, or 0.21%, to 23,833.38. The S&P 500 lost 0.82 points, or 0.03%, to 2,867.62 and the Nasdaq Composite added 84.88 points, or 0.96%, to 8,894.00.
The blue-chip Dow Jones index came under pressure from declines in oil giant Chevron Corp as crude prices fell. The S&P 500 energy sub-index dropped 2.2%.
General Motors Co jumped 5% after the automaker topped first-quarter profit expectations and outlined plans for a May 18 restart of most of its North American plants.
CVS Health Corp gained 2.3% after reporting a better-than-expected quarterly profit.
Oil slid below $30 a barrel as crude stocks rose and gasoline demand remained below normal seasonal levels, snapping a six-day winning streak for Brent futures that helped the global benchmark almost double in price from a 21-year low on April 22.
U.S. crude inventories rose by 4.6 million barrels in the week to May 1 to 532.2 million barrels, compared with analysts' expectations in a Reuters poll for a 7.8 million-barrel rise.
U.S. crude fell 5.9% to $23.11 per barrel and Brent was at $28.77, down 7.1% on the day.
Healthcare stocks rose in Europe on the back of better-than-expected quarterly results from Denmark's Novo Nordisk and German dialysis specialist Fresenius Medical Care.
"Earnings season is not great, but it's really the issue of the virus and the end of the lockdown, and sentiment towards that will push the market," said Francois Savary, chief investment officer at Swiss wealth manager Prime Partners.
China opened for the first time since Thursday and reversed early losses, sending its blue-chip index up 0.6%.
In a move seen by analysts as offering a olive branch to Washington amid the trade tensions, China's central bank set the yuan at a broadly neutral midpoint. The exchange rate has been a contentious point in China-U.S. ties.
U.S. President Donald Trump has repeatedly taken aim at China as the source of the pandemic and warned it would be held to account. On Tuesday, he urged China to be transparent about the origins of the coronavirus, which began in the Chinese city of Wuhan late last year.
Spot gold dropped 1.1% to $1,687.10 an ounce.
Additional reporting by Swati Pandey in Sydney; Editing by Bernadette Baum