* European stocks rebound after MSCI Asia ex-Japan ends down 0.3%
* U.S. bond yields steady after sharp dive, euro zone dips again
* Yen sets 5-month high vs struggling dollar
* Australian stocks rise after RBA rate cut
* Oil lurking near bear territory
* Asian stock markets: tmsnrt.rs/2zpUAr4
By Marc Jones
LONDON, June 4 (Reuters) - European stock markets and Wall Street futures clawed higher on Tuesday, though worries about a clampdown on the world's internet and social media giants and mounting recession jitters kept caution levels high.
Those nerves have pushed investors into top-rated government bonds and other safety plays in recent weeks, and there was little sign of a significant reversal as Europe reached mid-session.
With markets now pricing in as many as three U.S. rate cuts this year, benchmark 10-year U.S. Treasury yields steadied just above 2%, German yields stayed near record lows and the dollar skulked at a five-month low against Japan's yen.
The bounce by Europe's STOXX 600 was helped by a rise in Wall Street futures and by comments from Italian Prime Minister Giuseppe Conte that Rome had to abide by European Union budget rules for the time being, which cheered bank stocks.
However it was capped at the other end by a more than 1.4% drop in tech stocks after news the U.S. government was gearing up to investigate whether Amazon, Apple, Facebook and Google misused their market power.
That had wiped a combined $85 billion off Facebook and Google parent Alphabet's values and pushed New York's tech-heavy Nasdaq into 'correction' territory, having taken its losses over the last month past 10%.
"That (U.S. investigation) is currently weighing on stocks, but more importantly the market is increasingly pricing in the risk of recession," said Rabobank senior macro strategist Teeuwe Mevissen.
He called money market pricing of nearly three U.S. rate cuts before the end of the year "excessive", but Fed policy is in now in sharp focus.
Ratesetter James Bullard said on Monday a cut "may be warranted soon" given as the hostile trade rhetoric between the U.S. and China and pressure on growth.
Australia's central bank then cut its rates to a record low overnight, while on Thursday the European Central Bank will detail a fresh dump of cheap money and India is expected to lower rates too.
"It is possible that the current policy settings will be enough – that we just need to be patient. But it is also possible that the current policy settings will leave us short," Aussie central bank Governor Philip Lowe said after its cut.
That had nudged the Australian dollar down and Sydney stocks up. MSCI's broadest Asia-Pacific index ended fractionally lower though as 0.9% and 0.5% drops in Chinese bluechips and Hong Kong and a choppy day for Tokyo pegged it back.
The main focus remained on the bond market. U.S. Treasury yields ticked back to 2.1% but Monday's low of 2.06 - the lowest since September 2017 - was still within easy reach.
All this underlined the scramble to re-price Fed policy after the biggest two-day drop in U.S. two-year Treasury yields since the 2008 crash. The yield curve between three-month and 10-year debt was also still inverted by 25 basis points.
Adding to the rates rethink has been a recession-spooked recoil in world oil prices. Brent crude futures are now testing $60 per barrel for the first time in four months, having dropped roughly 20 percent since topping $75 at the end of April.
They were last down 1.3% at $60.48 per barrel, while U.S. crude was down 0.7% at $52.95. In contrast, safe-haven gold was up 0.1% at $1,326.47 per ounce, near three-month highs.
Back in the currency markets, the dollar index, which measures the greenback against a basket of currencies, slipped to 96.995, its weakest since April 18.
That was largely down to the yen gains and after the euro had hit a six-week high, though the shared currency was pushed back slightly to $1.1247 after weaker-than-expected euro zone consumer price inflation data.
"Risk aversion has also been seen with the yen carry trade unwinding as the markets comprehend that the U.S. technology containment strategy towards China is unlikely to reverse," analysts at Jefferies said in a note.
"In the short term, positioning has become so bearish that 'a ceasefire' could spark a risk rally," they said.
Additional Reporting by Andrew Galbraith in Shanghai; Editing by Larry King and Jan Harvey