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FOREX-Dollar trims gains as trade optimism fades, Aussie near lows post RBA cut

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* Graphic: World FX rates in 2019 http://tmsnrt.rs/2egbfVh

* RBA expected cuts rates to a record low

* Traders looking for clues on additional RBA cuts

* Greenback pares gains as caution on trade sets in (Adds RBA rate cut, yuan, analyst's comments)

TOKYO, July 2 (Reuters) - The dollar gave up gains on Tuesday as investors curbed earlier enthusiasm about U.S.-China trade progress while the Australian currency barely budged from recent lows after a central bank rate cut decision offered few clues about future easing.

The yuan also shed its early rise to trade lower on the day after U.S. President Donald Trump said any deal with China would need to be somewhat tilted in favour of the United States, suggesting negotiations may not proceed smoothly.

The U.S. dollar index against a basket of six major currencies earlier rose to its highest in a week but retreated as doubt set in about the resumption of U.S.-China efforts to resolve their trade war.

Market focus now shifts to Reserve Bank of Australia Governor Philip Lowe, who speaks to business leaders in the northern Australian city of Darwin at 0930 GMT, which could provide clues on how much further interest rates could fall.

"The tone from the RBA was not that pessimistic, which gives the impression they are somewhat reluctant to cut rates further," said Yukio Ishizuki, foreign exchange strategist at Daiwa Securities in Tokyo.

"Short positions in the Aussie were already so heavy. Now we're in a situation where the main risk is for the Aussie to be bought back."

The Australian dollar was up 0.3% at $0.6983 on Tuesday after slumping 0.9% on Monday, its biggest decline since April 24.

The RBA lowered interest rates by 25 basis points to a record low of 1.00%, matching economists' expectations. In a statement the central said it would lower rates again "if needed," a phrase some analysts took to mean an additional rate cut is less certain than before.

The U.S. dollar index was little changed at 96.790 on Tuesday having posted its biggest increase since March 7 on Monday on hopes Beijing and Washington were making headway in their trade negotiations.

The United States and China have already imposed tariffs of up to 25% on hundreds of billions of dollars of each other's goods in a dispute about China's trade practices that has lasted nearly a year.

The drawn out trade war has slowed global growth and pushed many central banks to cut interest rates to support their economies.

The offshore yuan gave up early gains to trade around 0.2% lower at 6.8690 versus the dollar, on course for its biggest daily decline in a week.

The global investor spotlight will move to U.S. non-farm payrolls data due on Friday, which economists expect to have risen by 160,000 in June, compared with a 75,000 increase in May.

However, analysts expect the dollar will struggle to make substantial gains given expectations the Federal Reserve will cut rates due to low inflation and worries about the U.S.-China trade war.

"It would be a mistake to view the rise in the dollar on Monday as the beginning of a broad-based rally," said Junichi Ishikawa, senior foreign exchange strategist at IG Securities in Tokyo.

"Treasury yields are capped around 2%, because there are still expectations for Fed rate cuts."

The euro briefly fell to an eight-day low of $1.1275 before trading little changed at $1.1289. The common currency fell 0.7% on Monday, its biggest-one day decline since March as disappointing economic data triggered a tumble in bond yields and boosted expectations for a European Central Bank rate cut. (Reporting by Stanley White; Editing by Sam Holmes)