In Australia, the ASX 200 closed largely flat, with the heavily weighted financial sector seeing a loss of 0.67 percent. Commonwealth Bank of Australia shares closed lower by 1.08 percent while Westpac Banking Corp's stock slipped by about 1.17 percent.
The Nikkei 225, on the other hand, saw gains of 0.29 percent to close at 23,940.26, with semiconductor manufacturer Tokyo Electron rising 2.42 percent.
Markets in Hong Kong and South Korea are closed for public holidays.
US-China trade war 'to get worse'
On Monday, China also released a white paper where it outlined Beijing's response to criticisms leveled against it by the U.S.
"China does not want a trade war, but it is not afraid of one and will fight one if necessary," Beijing said in the paper.
William Zarit, chairman of the American Chamber of Commerce in China, told CNBC on Tuesday that the conflict between Washington and Beijing is "going to get worse."
It's very possible that the Trump administration is going to start applying tariffs to "just about all" Chinese imports into the U.S., he added.
"It has to end with the two leaders coming up with some kind of accommodation," Zarit said.
Oil industry sees broad gains amid higher crude prices
Over in Japan, shares of refiner Cosmo Energy gained 3.72 percent by the end of the trading day, with Japan Petroleum Exploration also seeing gains of 1.48 percent.
In oil markets, prices continued ticking up in the afternoon of Asian trade. As of 2:57 p.m. HK/SIN, the global benchmark Brent crude futures rose 0.41 percent at $81.53 per barrel, while U.S. crude futures saw an increase of 0.31 percent at $72.30 per barrel.
On Monday, Brent crude prices spiked by more than 3 percent on the back of both Saudi Arabia and Russia ruling out any immediate increase in oil production.
Wall Street hit by political turmoil in U.S.
The moves stateside came on the back of reports that U.S. Deputy Attorney General Rod Rosenstein was on the verge of resigning or being fired. The White House later released a statement saying Rosenstein and President Donald Trump "will meet on Thursday" amid the conflicting media reports.
The U.S. dollar index, which tracks the greenback against a basket of currencies, was at 94.219 as of 2:55 p.m. HK/SIN, paring some of its earlier gains.
— Reuters contributed to this report.