 "Asian shares edge up on hopes of peaceful Ukraine solution")
There was no word, however, on movement of Russian forces that have effectively occupied much of Crimea.
European markets were likely to open higher, with financial spreadbetters predicting Britain's FTSE 100 to open 17 to 24 points higher, or as much as 0.4%, after dropping more than 100 points in the previous session. Germany's DAX was seen opening 6 to 24 points higher and France's CAC 40 14 to 18 points up.
"The lack of coherent response from the West may actually be providing some reassurance to traders that things won't escalate too far, with the reasoning being that the only thing the West will be firing towards Russia are harsh words," said Jonathan Sudaria, a dealer at London Capital Group, in a note to clients.
Russian President Vladimir Putin ordered troops that took part in military exercises this week to return to base, Russian news agencies quoted the Kremlin spokesman as saying on Tuesday. The exercises took place across western Russia, an area which borders Ukraine.
MSCI's broadest index of Asia-Pacific shares outside Japan added about 0.1% in afternoon trade.
The Nikkei ended up 0.5%, erasing earlier losses, on cautious bargain-hunting by investors hopeful that conflict will be avoided.
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"Investors seem to think that Russia will take Crimea and the West will impose sanctions on Moscow, but there will likely be no military clash," said Kyoya Okazawa, the head of global equity and commodity derivatives at BNP Paribas in Tokyo, who spoke before the news of the troop movement.
Russian financial markets rose early on Tuesday after plunging in the previous session on fears of a military confrontation. The rouble-denominated MICEX stock index added 2.7%.
Many investors had flocked to safe assets for fear of further escalation in tensions.
President Barack Obama said on Monday that Russia violated international law with its military intervention in Ukraine and warned that the US government would look at a series of economic and diplomatic sanctions to isolate Moscow.
The European Union also threatened unspecified "targeted measures" unless Russia returns its forces to their bases and opens talks with Ukraine's new government.
On Wall Street, the S&P 500 index lost 0.7%, to 1,845.73 as concerns over Ukraine overwhelmed generally upbeat economic data on the US economy.
Financial data firm Markit's gauge of US factory activity rose in February to its highest level since May 2010. Separately, the Institute for Supply Management said its index of US factory activity rose to 53.2 in February, topping expectations. Personal spending in January grew despite the harsh winter weather.
The data helped to underpin the dollar, with the dollar index rebounding from a two-month low hit on Friday to stand at 80.033, down about 0.1% on the day, but still well off Friday's low of 79.688.
The euro stood at $1.3752, up about 0.1%. While it remained off Friday's peak of $1.38255, it hit its session high of $1.3767 after news of the Russian troop withdrawal.
Investors' risk-averse mood had helped the yen, whose gains unravelled late in the session after the Russian developments. The yen fetched 101.78 yen to the dollar, down 0.3%, and moved away from a one-month high of 101.20 hit on Monday.
The Australian dollar erased initial gains after the Reserve Bank of Australia kept rates steady at a record low, as widely expected. The Aussie was flat on the day at $0.8936, not far from a one-month low of $0.8891 touched on Monday. It briefly rose to a session high of $0.8970.
Gold, another traditional safe-haven asset, skidded 0.8% late in the Asian session to $1,339.10 per ounce, after it touched a four-month high of $1,354.80 on Monday against the backdrop of the risk-averse mood.
Gains in US Treasuries sent the benchmark 10-year yield to a one-month low of 2.592% as prices rose on Monday, but the yield was last at 2.631% in Asia, up from its US close of 2.608%.
Retreating fears of possible armed conflict pushed down oil prices. US crude futures, which have been rising for weeks due to cold winter in many parts of the United States, were last at $104.22 per barrel, down about 0.7% on the day.
Three-month copper on the London Metal Exchange added 0.3% to $6,989.75 a tonne, after it skidded to a three-month low of $6,944 a tonne in the previous session.