* Asia stocks steady; China up, Japan, Australia down

* Spreadbetters see lower open for Europe

* Dollar on defensive as geopolitical woes nudge yields down

* Fed's Mester eyed for further cues for dollar

By Shinichi Saoshiro

TOKYO, Sept 24 (Reuters) - U.S. air strikes in Syria left Asian stock markets jaded on Wednesday, setting the stage for another soft session for European shares.

The dollar was kept in check after U.S. yields fell on geopolitical concerns and dovish statements by a Federal Reserve official.

Spreadbetters saw European equities starting lower, forecasting London's FTSE to open down as much as 0.3 percent, Germany's DAX and France's CAX down 0.2 percent each.

MSCI's broadest index of Asia-Pacific shares outside Japan initially fell to a four-month low after Wall Street's overnight slide, but managed to steady thanks to gains in Chinese shares.

The CSI300 of the leading Shanghai and Shenzhen A-share listings climbed 0.9 percent, while the Shanghai Composite Index was up 0.8 percent.

Space and defense stocks surged, with the industries enjoying support on hopes they would benefit from deepening reforms in state firms and from more government investment in defence.

On the other hand, Tokyo's Nikkei shed 0.3 percent and Australian shares lost 1.1 percent.

The United States and its Arab allies bombed militant groups in Syria for the first time on Tuesday, opening a new front amid shifting Middle East alliances and sapping demand for risk.

"If geopolitical concerns deepen, you can't expect Japanese markets alone to survive. The market could fall up to 10-15 percent at most," said Akiko Miyazaki, director of stocks at Barclays in Tokyo.

The air strikes in Syria also garnered demand for safe-haven government debt and pushed U.S. Treasury yields lower, in turn helping arrest the dollar's recent bull run.

The dollar was down 0.3 percent at 108.58 yen, after going as low as 108.25 yen overnight.

The greenback has been on the back foot after scaling a six-year high of 109.46 on Friday, receiving an additional knock after Japanese Prime Minister Shinzo Abe voiced concern about the economic impact from the currency's recent weakness.

The euro was little changed at $1.2851, limping away from the 14-month low of $1.2816 hit Monday.

Fed officials could offer more catalysts for currency markets, after Federal Reserve Bank of Minneapolis President Narayana Kocherlakota said on Tuesday the central bank can keep stimulating the U.S. economy because inflation is posing little threat - comments the markets perceived as dovish.

Federal Reserve Bank of Cleveland President Loretta Mester speaks later in the day.